It is 8am on a warm morning. Waterloo station in London is the city's busiest bicycle hire dock, and this is its busiest time of day. There are no bikes, of course. As quickly as vans unload cycles – 400 of them by 10am – the cycles disappear. Tutting rises from the queue. Then a man glides in, docks and strides off, and the first commuter in line has wheels. Today, wherever bike 15512 goes, we're going, too. We want to know how the cycle hire scheme is used, who rides the bikes, and why. What kind of London will one cycle see?

The man at the front of the queue is Mo Pour, 46, an IT contractor at Standard Chartered bank. Like many at Waterloo, he commutes to the City; the network's most popular journey is Waterloo to Stonecutter Street, near Ludgate Circus. Mo trills his bell in phrases of five behind walkers on their way to the South Bank. Heading towards Southwark, his long shadow tucks in beside him. And then Mo is crossing the Thames, knees sticking outwards, houndstooth jacket flapping.

"The Shard. St Paul's. Tower Bridge." He waves left and right, a jocular tour guide. London opens up on either side of him. But Mo's mind is on the other end of the day, when he will untether his own bike at Brookwood station in Surrey and swoop downhill past a river and lake towards his supper (fish on Wednesdays). Home in time to read his daughter a bedtime story.

Mo has got ahead of himself. It is only 8.30am. When he docks on Aldermanbury in the City, a wall of green windows cools the street. Other riders park and leave in a puff of chalk stripes. No one hires. The scheme is tidal. Unlike in Paris, where most users live and work in the city centre, in London traffic flows inwards in the morning and outwards at night. All day, 28 vans shift surplus bikes to empty stations, an unending battle against the current. The docking station fills. Half an hour passes.

Then a man in a caramel leather jacket appears. He cranks up the saddle several notches. "Sure, you can follow me," he says, "but it might be a little chaotic."

Kacper Chwialkowski, 26, from Warsaw, is hard to keep up with – "The bikes are free for only half an hour" (after paying £2 for the day) – and he is going all the way to Euston (he's a PhD student in statistics and computer science at University College London). He pedals the wrong way up Whitecross Street. Pedestrians jump back on to the kerb. The food market zips past in a blur of pleasant smells. When Kacper must slow down, at a red light he deems unarguable, he pedals backwards. No vehicle is too forbidding to undertake. He nips inside buses, threads between lorries – all habits he picked up here. "For sure not in Warsaw."

After 25 minutes, Kacper docks. "Why are you following one bike all the time?" he asks. "What's the reason for not changing the bike?"

"To see where it goes, who rides it."

Kacper's PhD thesis is "looking at the dependence between random processes". He suggests we save some trouble and get hold of the bike's log. "They've probably got the route encoded."

According to Transport for London's log, 15512 is an original phase one bike, meaning it has seen active service since July 2010. The number of daily journeys each bike makes varies according to the season: as few as two in winter, 18 or more in summer. So far today, 15512 has clocked up 7.2km.

At 10.30am two men appear: a big one with tattoos on a bike, the smaller one chasing him on foot. They appear to know each other, so that's OK. "All the bikes were broken at King's Cross," says Barry, the one on wheels. Chris, who has run beside him, checks out 15512.

Chris and Barry are on a pilgrimage. It is Barry's 45th birthday. "We were born in Chelsea," Chris says. "We moved out to Stevenage when we were young boys."

"Every birthday, his or mine, we come to London," Barry says.

They have taken the day off work – Chris is a media planner for the TV channel Dave, Barry is an engineering manager at GlaxoSmithKline – and got the train in from Hertfordshire. They want to dock the bikes at Covent Garden to keep within the 30-minute hire, but the rank is full. The scheme started with a ratio of 1.7 docking spaces for every cycle, as Paris has, but it has since moved up to 1.8. Space in the city centre is still tight. According to Nick Aldworth, general manager of the scheme: "The big push this year is to try to get more space in the middle." He says London has "roughly half the density of stocking points in the centre of town than New York", just one reason the vans are so busy. But none of that helps Chris and Barry today. They are soon sailing over Waterloo bridge. There goes the Shard again. Barry and Chris cross Mo's route.

"The problem was, after we moved, my dad was taken ill with cancer," Barry says. The traffic is heaving past on York Road. "So me and my brother moved back to London to live with our nan and grandad while Dad was being treated. We stayed a year. He had cerebral lymphoma. He did well – he survived it for four years." We collapse to single file as a freight truck overtakes. Lambeth Palace flies past.

So this is a sentimental journey, then?

"It's purely that," Chris says.

"Mum was a young bride," Barry says as they dock their bikes at Fire Brigade Pier in Vauxhall. "When she moved out to Hertfordshire, it was a wrench to be away from her mum and dad. Every weekend we came back. Pie and mash." Today, lunch is in Tooting. "'Cause that's the nearest pie and mash shop we know."

The boys – that is what they look like, with their close-cropped hair and matching camouflage shorts – hire new bikes. They ride off, two abreast, standing on pedals, towards their "old stamping ground" of the Winstanley estate.

The Thames flops over traffic cones and bus stop signs. A young guy docks a bike and sprints away. Big Ben chimes 11.30. The sprinting guy sprints back.

He adjusts the saddle on 15512. How fast does he cycle? "Fast."

Toon Bakkeren, 19, may have been  born on a bike. To indicate, he extends two fingers. Perhaps he has changed gears after Chris, because the pedals fly round. Lambeth bridge flattens beneath him. "That was just a small chore," he says, nodding back to the other side of the river. "Remax real estate company. I'm an intern there. They sent me to get the mail." He is 19, from Breda in the Netherlands, and funding his stay himself. "It's quite painful." He docks after eight minutes, outside the Queen Mother Sports Centre in Victoria.

Another van pulls up and takes away surplus bikes. It's the driver from Waterloo again, on his 14th job of the day. He leaves bike 15512, which has now travelled 13.1km, and Elisabeth Kriegsmann checks it out.

"I live in Stockwell," she says. "That's where I'm going. I'm picking up my boyfriend and getting him out in the park." She has finished her shift at Riverbank Park Plaza hotel. "Meetings and events," she says. Ten days in a row. "So today is hanging out." She heads for Vauxhall bridge and the bike crosses the Thames for a fourth time.

Traffic streams past on Wandsworth Road until Elisabeth turns left and there is silence. "The last bit of the journey," she says. "Quiet corner in London."

She docks the bike. It is troublingly quiet. "Sometimes it's really busy," Elisabeth says. "If not, I'll take the bike in an hour or so."

Pop, pop, pop comes a Pizza Hut delivery bike. A bird sings, its song like the opening of a tin can. It is nearly 3pm. Two hours have passed in this strange pastoral. So much for Elisabeth and her plan to go to the park. Maybe she snuck out the back door. All is still. Just the shadows sharpen and soften. A Papa John's moped crawls past. No one has hired bike 15512. We will have to take it to get lunch. We push it to Vauxhall tube, a 10-minute walk, and lock it to railings. After lunch, we wheel it round the corner to the docking station. If the chain is to be broken, this seems a small disconnection.

Within minutes, a woman approaches, smiling. She looks familiar, yet different. Elisabeth! In a new outfit. Wasn't she going to hire another bike near her house?

"I was, but then I remembered I had an appointment." It is possible Elisabeth hoped to ride to the park alone with her boyfriend, but fate has brought her back to bike 15512.

She lowers the saddle – too high for the skirt – and heads back over Vauxhall bridge. She is from near Nuremberg in Bavaria. Like Toon, she came to London to work as an intern. "I wanted to stay longer, so I started working in a hotel. I'd like to study here, but it's really expensive. I will probably go back to Germany in the autumn."

Elisabeth sails past Buckingham Palace and into Green Park. Will her boyfriend, Steven, go to Germany? "We're not quite sure," she says.

There is a shortage of cycles at Green Park, and at 5.13pm Barbara Nowakowska, 54, from Warsaw, becomes 15512's next customer. Her conference over – she is managing director of a Polish private equity and venture capital firm – she wants to enjoy the sun, see the park. "I love Boris bikes," she says.

She wobbles along the pavement and every few pedals freewheels. Hyde Park, the third busiest dock after Waterloo and King's Cross, is full of hire bikes. After 30 minutes, Barbara swaps bikes on Bayswater Road and Humayun Kobir shunts bike 15512's saddle down to its lowest setting. He has come into London from Luton, "for visa purposes". From the bus, he looked into Hyde Park and saw all the cyclists, thought he'd like a go. He sets off in heavy leather jacket with fur-lined hood, even though the sun is scorching. At weekends, when 70% of the scheme's users are non-members (the figures are inverted in the week), Hyde Park to Hyde Park is the most popular journey.

Humayun must be hot in that jacket. "Yeah, yeah," he says. By the Serpentine we stop for an ice-cream, Humayun's first 99. Back in Bangladesh, he is "a senior information officer at the Ministry of Information", but here he is "doing an MSc in project management". Humayun and his wife, also a civil servant, came to the University of Bedfordshire on government scholarships after collecting International English Language Testing System points. "This is somewhat difficult for me," he says, nodding at the 99.

With Humayun in the saddle, 15512 skirts the Serpentine where another hire bike and rider have fallen in the water. "There's a lot of system here that my country can pick up," he says. "Better technology. Better lifestyle. If I park a car, I can pay the machine. And in the library I can take a book just using my card. Although my country is starting to do better with digital, it is lagging behind. I am only one person, and it is difficult for me to do something about it, but still, the UK has influenced me a lot."

He is torn between returning the bike and taking photos. The photos win, because when he docks near Marble Arch, he has had 15512 for 50 minutes – his 2.2km bring the total to 23.1km. Tourists cluster around the bike; endless laps of the park loom. Every other bike is showing its fault light. There are more than 10,000 bikes in all. Each day around 200 bicycles – made by Canadian company Devinci – are serviced at the Islington depot of Serco, the outsourcing company that manages the scheme. Every bike is seen at least twice a year, but the system appears to have fallen down here. Fortunately no one wants to hire a lone bike. Everyone has friends.

By now, Mo, our first rider, is flying past the lake in Brookwood, nearly home. Kacper is leaving university. In Stevenage, Barry and Chris's families are waiting to sing happy birthday.

Eventually, Tony Wong and David Wang turn up. They are both 20-year-old students from Beijing, on their way to dinner at Yauatcha in Soho. Tony is imagining black pepper deer dim sum, while he rolls along on 15512, steering with his left hand, right arm hanging loose.

It is 7.30pm when Tony and David dock in Soho. Alan Waller, a sustainability analyst, says he is happy to be followed to the gym, provided the bikes are not referred to as "Boris bikes". "The scheme was secured under Ken Livingstone," he says. "Boris took all the credit. I try not to call them Boris bikes."

Alan has barely docked, just off Tottenham Court Road, when Silvia Langford, 52, finds that 15512 is again the only functioning bike. By now it is nearly 8pm. Silvia, who runs her own publishing company, is heading home to Shoreditch. She wriggles through the quiet roads of Bloomsbury and soon hits Guilford Street, where Kacper crossed earlier.

At a red light, she says, "With any luck, my partner will have cooked for me." At another, she says, "It's quite unusual having someone to talk to. No one ever talks to anyone on a bike." It is dark when Silvia docks near Columbia Road, and 15512 has travelled 30.9km since Mo took it out at Waterloo station this morning. A couple of minutes later its lights stop flashing, as if the bike has shut its eyes. Silvia returns with her business card and says, "If you want an update, my partner's watching the football."

It's 9.30pm. Alan is hiring one last bike for his ride home after an intensive leg workout. Time to call it a night. By 9am tomorrow 15512 will be gone again. But that's another day.

• This article was amended on 29 July 2014 to correct the spelling of Nick Aldworth's surname, from Oldworth as an earlier version said.