Wait, so what about the Inbetweeners? Wasn't that a BBC show that got turned into a film fairly quickly?
As said above, Inbetweeners is a channel 4 show. Channel 4 gets a few public grants here and there, but is not taxpayer funded. They have ads, etc. A lot of non-BBC British shows end up being taken Worldwide by the BBC, though, so a lot of people know shows like The Inbetweeners and Peep Show as 'BBC shows'. That's part of the BBC's taxpayer-funded remit - to take the best of British, even non-BBC stuff, and spread it to the world as a way of showcasing British culture.
The Thick of It ---> In The Loop?
In The Loop is a weird old case. I think. It's actually a UK Film Council movie - the BBC collaborated and gave them a ton of money and production aid, but the main producers are actually the UK Film Council (which was axed a few years ago). It also had a ton over overseas funding - some Canadian, some American, some French. The UK Film Council was funded by National Lottery grants and stuff. It's a weird case. Armando - the writer/creator - retains the rights to all his shows & characters, though, like Alan Partridge, who has been used for corporate promotions and done private show as well as BBC stuff. So he could've taken In The Loop to whoever he liked for funding and used Tucker and stuff. The BBC just stepped up and offered them cash from their very limited film budget. They recently helped him to fund that new Partridge movie, too.
Aside from all that, In The Loop is completely unrelated to The Thick of It, really. Malcolm and Jamie are in it, but all the other actors play different characters (a brief appearance by Tucker's Secretary & Angela Heaney aside) and the Prime Minister referenced in the script/cut sequences is a different one to The Thick of It. So it's an alternate universe, in a sense.
That said, this stuff is getting easier. Like, where in years past Clarkson/Hammond/etc used to do non Top Gear DVDs that were Top Gear in all but name for Christmas, this year they've actually got a Top Gear-branded DVD out which is basically an exclusive episode. Things like that, and the fact the BBC Worldwide bought a huge stake in Clarkson's Top Gear production company and stuff has aided that. So the BBC is becoming more adaptable, slowly, but the rules are still tight and difficult. Especially now, where elements of the British government would love to strip them of all funding and make them a profit-driven organization - it's more important for them to tread carefully than ever.