Just to ask Is the UKTV name staying since BBC now controls it all? Also what's the deal with the UKTV stings on the Food network?
It's important to be clear when using 'BBC' to discuss this stuff.
UKTV is now a part of BBC Studios (which is the combined company formed by BBC Worldwide and the non-Sport, non-News, non-Kids, non-National/Regional arms of BBC Productions). This itself is a commercial subsidiary of the BBC - like BBC Studioworks. In some ways - think of BBC Studios as an independent production company that happens to be owned by the BBC (like Endemol/Shine or Tinopolis)
There is a significant separation between BBC Public Service/BBC Content and BBC Studios (and thus UKTV). I don't think BBC Studios would be allowed to rebrand UKTV services as 'BBC' without doing a deal with their owner, BBC Public Service, and I doubt BBC Public Service would want (or Ofcom allow?) BBC branded services to carry commercial advertising within the UK.