I doubt the BBC will drop him.
He makes them a lot of money, even if he is controversial. Same reason ITV really enjoys bringing Katie Hopkins on tv.
If they do drop him, I'm sure ITV or Sky will pick him up ridiculously quick. Especially if there's a chance that Hammond and May will follow when their contracts are up.
he also supposedly murmured the word "monday" according to another news post.
This really was a no-issue.
He recited a nursery rhyme, which historically used to use the word 'monday' in it.
And they then-and-there decided the line wasn't good to use, so they replaced it. They never once aired the word monday.