In Australia we have a show called "Utopia" that does fill this gap reasonably well. Australian politics are close enough to the UK that it'd probably translate well enough to be enjoyable.

I've heard many government workers say that it's funny but they can't watch it, as it's so accurate it's depressing.

Just seconding a vote for Utopia (2014), and also its (better!) predecessor The Hollowmen (2008).