For example, when the Butchers sit down in the living room, why not have the EE theme tune playing on the telly for a few seconds?

Would this add realism or make the show a bit of a mockery?

I doubt this has ever happened before, unless for comic relief, or something.

It would make a mockery. Tptb do try hard to make the show realistic and that's why they cover headline news and why Dot made reference to watching corrie.

I agree with Scrabbler to be honest. I just think myself it'd make a bit of a mockery out off the show and it's be just plain strange. Maybe something subtle like that time Dierdre and Ken were watching Dancing On Ice on the same series the man who played Ashley in Corrie was on (I lurked here back then and remembered that was a well loved scene)would be okay but somebody who's a character on Eastenders directly sitting down and watching an episode would be unexplainable.

This makes me wonder though, in Eastenders world what on earth is on at the time that it should be on, since they have been seen to be watching the BBC quite frequently

The fictional soap Westenders, set in the west end of London focusing on a fictional theatre and bars in the west end.

Doctor Who nearly did this once.

https://www.youtube.com/watch?v=7mLxvL76yG4

yes there is no way the residents of Albert Square would be watching EastEnders, the BBC of the EEverse, would be airing another soap, WestEnders being as gooda choice as any.

Is the gameshow Cat & Mouse still running in the EastEnders universe?