I had a quick look at the home office site here:

https://www.gov.uk/check-uk-visa/y/u...ewhere_else/no

It says that she won’t need a visa if transiting.

As for the bags, it will depend on there being an inter line agreement between those two airlines. You may be better off checking in a different forum as this is dedicated to British Airways. I’ll ask for this thread to me moved somewhere that’s more appropriate for the questions.