Travel Bug : Just call BA and ask them what their intentions are.
My experience taught me that you have to follow them all the time, keep calling as the left hand doesn't know what the right is doing.
Phone them, ask them for a start date. Tell them you haven't received anything in the post yet and ask them to confirm that the address they have is correct. When I was in your place, they had sent paperwork to my address in Italy - which I had never given to them but was the address of a previous application I had sent in 2001!!
Call them (recruiting or HR department) and I am sure it will be easier for you to decide what to do.
As for joining another airline in the meantime....do you really want to go through training (especially SEP) just to have to forget it quickly and start with BA after a few weeks?
Good Luck
FBW