When Rachel Reeves became chancellor in 2024, she said it felt like “smashing one of the last glass ceilings in politics”. But the presence in her office toilet at the Treasury of a urinal, thought to have been used by Winston Churchill, proved that there are some obstacles that can’t be overturned. Reeves would just have to tolerate this symbol that puts the “pee” in patriarchy.

For Churchill’s Urinal, writer and actor Rosie Holt takes a frenzied approach to the problem. Her sustained mania, familiar from her viral pandemic videos as a Tory backbencher, suggests a one-person Thick of It, though there are occasional interjections from Michael Lambourne as the taunting Churchillian voice – and face – of the urinal: WC played by the WC. Talk about toilet humour.

In a parallel Rachel Reeves-less reality, albeit one where Keir Starmer remains PM amid talk of “Team Wes” and “Team Andy”, Holt plays a chancellor faced with the same conundrum as Reeves. Her initial objections to the urinal are wilfully misinterpreted by the rightwing media as a repudiation of Churchill’s values. When she drops an accidental on-air truth-bomb, describing the former PM as a “turd”, it only inflames the controversy. By the second half of Daniel Clarkson’s production, the country is in uproar and the now-deranged chancellor is feeding vodka to the urinal. “Come to Winnie,” it coos.

Holt’s script, with additional material by Stewart Lee, makes salient points about how violent online abuse is now an occupational hazard for female politicians. The show is also fleetingly incisive on the matter of pro-Churchill myth-making, with a Darkest Hour gag the standout. These are anomalies, though, in a shambolic evening which too often relies on name-checks (Michael Fabricant! Isabel Oakeshott! John Nettles!) in lieu of insights or laughs.

It is the play’s misfortune to be opening just as Saturday Night Live UK has brought vitality back to political tomfoolery. Even if that were not the case, one would have to conclude that multiple jokes about Michael Gove, not to mention a would-be farcical telephonic mix-up that seems to go on longer than the last Tory government, amount to a clear case of taking the piss.

• At King’s Head theatre, London, until 6 June