BA should interline the baggage if it's on the same ticket. It's not their policy to do it if it's on separate tickets but technically they could.

AC's policy is to do it regardless of whether it's the same ticket or separate (if they have an interline agreement with the other airlines which of course they do with BA).

Note that when connecting *to* BA they want you to show the bag tag to a BA agent to make sure your bag is in their system. I'm not sure why their systems require this to be done manually but my experience is BA's IT is all done manually.