You don't need to ask permission from the BBC/ITV/Channel 4/Sky to use their logo for the purposes of a mock because it will probably fall under fair use and as long as you don't gain financially from it, you should be okay.
We don't really have fair-use in the UK (not like they do in the US where you can claim virtually anything is fair use as a reason to ignore copyright), and unauthorised use of broadcasters logos is not only breaking copyright rules, but trademarking ones too.
On the other hand, the broadcasters aren't going to care about it if you're just a school kid making stuff in your spare time -
you do something that brings their name into disrepute. Not making money wouldn't be a factor if you got something to go viral which caused lawyers to become aware of your work.