Still no ticket number, but I did, just now, get this:

Your e-ticket receipt
Dear Skip, Booking reference: 7*****
Thank you for booking with British Airways.
Ticket Type: e-ticket
This is your e-ticket receipt. Your ticket is held in our systems, you will not receive a paper ticket for your booking.
If your itinerary includes an American Airlines flight, we recommend you contact us to pre-assign seats.
What to do next
It went on with various caveats and instructions re the booking. Do E tickets get numbers? Or is the 7 character booking reference it? It is the same as the "PNR" I got from the booking tool with the instruction to call BA to book.
I'm going to PIT to the Steelers game this weekend, so I pulled up my booking on AA, and there is no number, just the 6 letter record locator. So I'm still perhaps over optimistically hopeful, lol.
And also hopeful I can select my seats with a US BA phone number!!