Maybe it wasn't ticketed because the rebooking violated the rules: different departure point, FC rather than business class, and use of a nonpartner carrier.

The second and third items are pretty easy to fix: AA books you in business class, probably using AA connecting to BA at LHR. The first is tougher because you've already reasonably made plans to be around LAX on the day of departure, so IMO it's unfair to "fix" this so long after the flights were rebooked.

In addition, since you say that someone from AA called you to rebook, it should have been an experienced agent who is an expert in the rules in general and also the rules that specifically applied to this situation. I wouldn't have expected that person to make a booking that wasn't permitted or would not be ticketed.