Given Boris Johnson’s daily catalogue of calamities, the time does not seem especially right for satire on the bygone deeds and misdeeds of Tony Blair. But this musical comes from the childish mind of Harry Hill, so the politics almost always plays second fiddle to the silly.

Yet the result is wildly inconsistent, occasionally making serious, even sombre, points about the cost of war which sit awkwardly against the broad stupidity and crass jokes that otherwise reign. 

Subtle it is not, with familiar cartoonish stereotypes such as the ‘look you boyo’ Welshness of Neil Kinnock or the hard-drinking Northern pugnacity of John Prescott. However, Alastair Campbell is portrayed as a Scot in kilt, bagpipes and tam o’shanter – which is odd given that he’s from Yorkshire. Actual Scots Gordon Brown and Blair himself are spared such a fate.

A pantomime messiness is Hill’s brand, of course, and his trademark knockabout energy propels this charabanc. There are some great jokes – an addendum to Princess Diana’s ’three of us in this marriage’ bombshell is particularly strong – but many more corny, puerile punchlines. ‘He put the "dick" in dictator,’ for instance.

Generally, the more preposterously tasteless the show gets, the better. The image of Saddam Hussein as a wisecracking Groucho Marx figure is irresistible. And of the lyrically dextrous songs, co-written with regular musical collaborator Steve Brown, the stand-outs are the most over-the-top moments when Osama Bin Laden sings Kill The Infidels or the showstopper which reminds us the whole world is run by assholes. (Though even Blair’s fiercest critics might think lumping him in with Stalin and Mao Zedong is a bit harsh)

Hill sees Blair’s arc as operatic: the rock-and-roll idealist who ended up a law-breaking warmonger through hubris. With an empty smile and lack of convictions, he’s depicted as only getting into politics to meet ‘Mick Jaggers’, rising through the ranks via a machiavellian Mandelson, developing a Messiah complex on the back of his successful handling of Diana’s death, and getting dewy-eyed over the gung-ho cowboy George W Bush.

It’s probably a mistake to show Blair’s entire life story as the first half gets bogged down in his uneventful rise to power (‘Blimey, we’re getting through it, aren’t we?’ comes one of many fourth-wall-breaking asides) and the show really only picks up after the interval when we get to the nub of his moral downfall.

Every headline-grabbing incident is covered, however briskly. Bill Clinton runs across the stage just to say: ‘I did not have sexual relations with that woman’ apropos of nothing. Why? It’s not a joke, just an acknowledgement of a quote we all recognise.

Stand-up Charlie Baker does the best he can with the inevitably two-dimensional portrayal of Blair. Until he finds power, there’s a touch of the Anthony Aloysius St John Hancock about his Anthony Charles Lynton Blair, a bit deluded and naive. Baker performs with the required vim and with a glint his eye, but doesn’t completely drown in the sloppy shenanigans, keeping his facial clowning relatively restrained.  

The cast – all dressed in New Labour uniform of dark suit and black tie – are strong throughout. Gary Trainor is impressive as the dour, perpetually shafted Brown, Howard Samuels is an arch panto villain, and Holly Sumpton grabs attention as a seductive Lady Macbeth. Sorry, Cherie Blair. Delicacy is not required and for the most part they mug their way through skits so vaudevillian that there’s an actual slap stick employed.

Yet the lack of sophistication – as well as the uneven pace – is the show’s undoing. For rarely is the script outrageous enough to overcome its slapdash student revue feel, and the impression that Hill is lightly pawing at easy, now irrelevant targets with jokes we essentially already know.

• Tony! [The Tony Blair Rock Opera] is on at the Park Theatre, Finsbury Park, London, until July 9.