No you'll have the same flexibility.

I would definitely try booking online, as my (and others') experience with the N. American BA RTW desk is miserable. If your first segment is BA over the Atlantic, be prepared for a much higher final cost than if you used AA, as a ticket issued by BA will undoubtedly charge more for fuel surcharges than if AA was the issuer.

You might try a little experiment with this - go online and book with BA as your first segment (e.g. LAX/SFO/PHX - LHR) and see the bottom line, especially the taxes and fees breakout (but don't pay for it) then try the same thing using something like LAX/SFO/PHX - ORD - LHR with AA as the first segment, and price it that way, which would force an AA issuance of the ticket. Don't know for sure, but I expect the bottom line would be quite a lot less.