As per their public statement in WSJ article today I believe, OW do not block any awards from each other. If it's available on one, it's available on all. I have found in experience this has always been the case too, though sometimes you have to be a bit insistent and ask the agent to check with ticketing.

AA is tough to get multiple awards on. BA is very good I have seen as many as 10 F seats on a single flight.