So apparently you're not allowed to play this on the radio. And reading the Broadcasting Act, its easy to see why: it "appears to encourage or persuade voters not to vote for a political party or the election of any person at an election", which makes it an election programme, which means it cannot be broadcast unless as part of a party's broadcasting allocation.
...which is a perverse result. What the Electoral Commission is telling us is that insofar as it can be viewed as persuasive, broadcast political satire is illegal. And I'm not sure that that really fits with their BORA obligation to interpret the law so as to be consistent with the right to freedom of expression. OTOH, it might explain why political comedy has disappeared from our television screens: because the politicians have outlawed it.
(And in case you're wondering: while apparently an "election programme" under the Broadcasting Act, that video isn't an election advertisement under the Electoral Act, for the simple reason that it is very obviously not an ad. So I don't think there's any problem posting it on the web, or performing it live outside Parliament).
Update: Update link to a non-FaceSuck source.