The BBC is a weird corporation created by royal charter. But it's not a part of the UK government, and nor are works created by the BBC copyright-free (as is the case in the US for something like NASA).

UK government publications aren't copyright-free either. In fact they manage to be worse than copyrighted, at least for works created before 1988 (some of which are perpetually copyrighted, others until 2040, others for 125 years, it's a big mess). https://en.wikipedia.org/wiki/Crown_copyright#United_Kingdom