All worked out fine.
Our BA flight arrived early, passport/ Immigration was a breeze under 10 minutes, bags came out 20 minutes after that. Easy drop off in baggage area to AA.
They wouldn’t let me thorough priority security since apparently it was only for TSA pre, even though the board says I was eligible. So that took an extra 25 minutes.
So all in all less than one hour from landing to sitting and drinking wine in the Greenwich lounge.
Thanks for all your help.