Ha! This is exactly what happened to me in May... I booked a ticket YUL-LHR via JFK, with the YUL-JFK sector on AA. I, too, am a long-standing BA Exec Club member, and have made countless Avios redemptions online, never with a problem before. And that has included some redemptions on AA, not just BA.
But this time I got to the AA check in counter at YUL (having not been able to check in online), and was told that the reservation had not been ticketed. Luckily, in my case, a really lovely and gutsy AA ground staff member spent 40 mins on the phone with BA, and in the nick of time, BA processed the reservation and put in a ticket number, at which point I was almost frog-marched through YUL by this wonderful AA staffer, and made the flight. (It helped I was only on hand luggage!)
But as much as the AA ground staff were great at YUL, BA were terrible on the phone. She first called the main BA number in YUL, and they were not open (this was 5pm, before BA check-in for the non-stop YUL/LHR). So she phoned the main North America BA number. They wouldn't deal with it, as it was Avios not cash, and gave her another number. That number was a number in Europe (at the time, my BAEC account was in Spain) - where they would not be coming into the office for another ten hours, given the time difference.
It was only really persistent and good quality customer service that finally found someone half competent in BA to get on the case - and even then it took 25 minutes of holding before they ticketed the reservation.
And let me just say once again - I had gone through the same Avios booking process on ba.com that I have done, probably at least fifty times in the last ten years. It was confirmed in just the same manner. Nothing in terms of what I saw on the website and was emailed gave me any reason at all to think that this booking was somehow messed up.
So, in the light of the original posting, there is clearly a major glitch somewhere....