As a comic storyteller, Mike Birbiglia is a finely tuned, perfectly engineered Rolls-Royce. As a human, he’s a clapped-out 1978 Honda Civic, struggling to stay on the road as essential parts splutter away.

The indignities of an imperfect body are a comedy staple, but the genial 45-year-old has a rare talent for weaving his anecdotes into a broader, compelling narrative while keeping the jokes coming at quite a lick.

As already proven in shows such as Sleepwalk With Me and My Girlfriend’s Boyfriend, Birbiglia offers a masterclass in ensuring not a word is wasted in his stand-up. Every line sets a mood or triggers a laugh in a story with a solid emotional core and a pleasing circularity, far richer than a cheap callback.

The Old Man And The Pool – arriving in the West End after a successful Broadway run – starts at a 2017 health check-up, which threw up concerns about his breathing and the condition of his heart (and, later, he’d discover type 2 diabetes, too). When the comic asks if he’s having a cardiac incident there and then, his doctor can only respond with the far-from reassuring: ‘I don’t *think* so.’ The troubling prognosis inspires a reluctant health kick, taking up swimming at his local YMCA, despite such pools triggering unwelcome childhood memories. 

Health humiliations have featured in Birbiglia’s previous shows. Fans will know of his life-threatening sleepwalks, which require him to be cocooned in a virtual straitjacket each night, and of his teenage bladder cancer, successfully treated but still a lifelong concern. A cystoscopy makes an appearance in this show, another much-mined occurrence for middle-aged male comics, but not dwelt upon for too long.

Meanwhile, other detours take in the likes of his ill-fated ‘career’ with the high school wrestling​ team or tackling mildew in his Brooklyn apartment, alongside fragments of wider observational comedy about Airbnbs or a rich guy’s doorbell, which is acted out masterfully. But every mini-routine is anchored in the story, and loops back into it seamlessly at the end.

The staging subtly enhances his storytelling. Lights dim slightly for reveries and come up for the present-day – especially when signifying a harsh clinical reality. On a simple, wave-shaped backdrop, a grid of squares sometimes represents the tiles of a swimming pool, sometimes graph paper to illustrate some medical test or other – or maybe just echoing the technical rigours of his scientifically precise writing.

That doesn’t overshadow the personal core of the show, that Birbiglia’s health problems have serious consequences. Both his father and grandfather suffered heart attacks at 56, which preys on the comedian’s mind. What would become of his young daughter should something happen to him? What joys of her growing up would he miss?

These are weighty worries that Birbiglia doesn’t shy away from, but nor does he over-exploit, never laying on emotions with a metaphorical trowel. The Birbiglias are not an ‘I love you’ family – another key strand of the story – but it means he’s not going to start gushing now. But the spectre of death adds pertinence to every story, while his recreations of bedtimes with little Oona are full of honest tenderness.

For most of the 80-minute show, this ultimate Everyman is bright-eyed as he tells the anecdotes, a twinkle of satisfied glee at what he has to impart. Even though Birbiglia is the ever-relatable loser on almost every beat, his ailing body ever-ready to bring him down a peg, the joys of living are what keeps him – and the show – so buoyant.

We remain rapt throughout, and Birbiglia’s command of the audience comes to a head when he tries to hold a moment of silence for a man who died in the pool. Of course, it’s impossible – no laugh is more potent than a forbidden one – and he forlornly tries to quell the smattering of giggles that erupt, Whac-A-Mole-like, in various corners of the Wyndham’s Theatre. 

It’s a typically virtuosic display from a comic who knows how to eke every laugh out of what could be a gentle story, with expert phrase-making, sharply efficient writing and a thoroughly engaging nice-guy performance, all wrapped around significant but skilfully underplayed themes.

• Mike Birbiglia: The Old Man And The Pool is at the Wyndham’s Theatre, London, until October 7. Tickets.