Nick Hornby’s 2014 novel Funny Girl was about a woman with the mind of a comedy nerd and the looks of a pin-up model: Barbara Parker may have been crowned Miss Blackpool, but she was desperate to escape her home town and follow in the footsteps of her idol, Lucille Ball. It’s no spoiler to reveal that she becomes a star – albeit in the musty world of mid-century British sitcom rather than its glamorous US counterpart.

Funny Woman, the Sky adaptation of that novel, stripped away this fixation with comedy. In series one, Parker – who changes her name to Sophie Straw at the behest of a sleazy agent (Rupert Everett, in an off-puttingly hammy performance) – moves to London simply to “be someone”, ending up a TV sitcom star mainly by chance. It was slightly disappointing to see the book’s USP played down in favour of something more generic, but – ironically – Funny Woman did improve on its source material in one crucial way. In the book, we are told Sophie is hilarious, although we rarely witness it. Gemma Arterton actually managed to imbue our deceptively daffy protagonist with quick wit and exemplary comic timing.

Now Sophie is back, and after the success of her odd couple sitcom, Barbara (and Jim), she is fast approaching national treasure status. When she switches on the Blackpool Illuminations, she has the audience in the palm of her hand. She sidesteps into music, too, with a TV special called Sophie Sings. She is in love with her buttoned-up producer Dennis (Arsher Ali), who – obviously – loves her back. In fact, everyone wants a piece of her: at a boho party, she is even approached by a Hollywood agent.

While series one traded charmingly on the culture-clash comedy of a working-class naïf finding her groove in the hallowed halls of the BBC, Sophie’s newfound self-assurance and celebrity means this second outing struggles to find much conflict, class-based or otherwise, to propel the plot along. It tries: the return of the mother who abandoned Sophie as a child results in many a mournful stare. Her relationship with Dennis is complicated by a delayed divorce in a storyline mired in dreary legal technicalities. After a rather unbelievable mix-up with TV bosses, Sophie ends up making a dud sitcom without Barbara (and Jim) writers Bill and Tony, but it’s so short-lived it barely tarnishes her reputation.

Funny Woman might not have any compelling problems, but it does have one fundamental flaw: Sophie herself has become tediously perfect. Not only is she unbelievably stunning – each scene offers an opportunity for her to don another eye-catching outfit – but she seems to have a faultless character, too. She never betrays any irritation with the noncommittal Dennis; she does her damnedest to inject life into even the weakest comic material; she proves to be a loyal friend by refusing to swap her shared bedsit for a swankier pad. In the book, Sophie is coldly ambitious. Here, she helps other women through the hole she has neatly cut out of the glass ceiling, casting a random barmaid (an underused Roisin Conaty) in another sitcom, a two-hander called Flat Birds.

Perfection is never funny, and this series is lighter on laughs than the first. When brainstorming titles for Flat Birds, Sophie’s suggestion is Two Funny Fannies. “You should never have funny in the title,” says Tony, in a meta aside. “It’s just asking for a kicking.” That doesn’t apply to this series. The retro sitcoms the team produce are clearly not meant to tickle modern viewers (I understand why: it would be very difficult to create a plausibly classic comedy that could also amuse 21st-century audiences), but even off set, Funny Woman never strains for laughs.

Instead, the overriding effect is gentle warmth – and on an ambient level, the world of Funny Woman remains an inviting one. The sets are a vision of cosy nostalgia. The rehearsal-room camaraderie is infectious. It has a feelgood sentimentality, too: the show is, perhaps anachronistically, fixated on progressive values. When Bill is arrested for soliciting sex in a men’s toilet, it is Sophie’s witness-box testament to his friendship and humanity that helps him dodge jail time (of course it is). Sophie’s flatmate, gloriously down-to-earth Croydonite shop girl Marj (the brilliant Alexa Davies), becomes involved with the women’s movement, while her friend Diane fights back against racist double-standards at work.

By the end of series two, Funny Woman has moved far beyond the scope of the book that inspired it and judging by the final episode’s cliffhanger, a third series is on the cards. But rather than a compelling character study, it now resembles a classy soap: the storylines driving the action along feel superficial and tacked on. The result is a comedy-drama as obviously contrived as the old-school sitcoms it eulogises.