They release them at midnight-ish central time. I will say you need to be watching the BA website to make sure they are still available by the time that AA releases them. JAL and BA open their calendars 355 days in advance and AA only 330 (or something like that).
Dates of when I booked our trip for reference:
AA Booking
6/17/23 LAX to KIX booked on 7/21/22 at 1AM
BA Booking
7/5/22 HND to DFW booked on 7/21/22 at 7PM
By watching the BA website I knew that there were still 4 J tickets available for me to book via AA.
But the flights from HND to DFW never lasted long enough to book via AA, they were usually gone within a few days. JAL is weird, they will release 2 J tickets, then 3/4 days later will release another 2 to 3 in J. Sometimes those 2 J tickets were booked before the other 2 tickets in J a released, specially the flights to/from HND. the flights to/from KIX usually made it to AA.