Arriving into TBIT last month (c2000) there was no wait for immigration. Lots of machines available for ESTA travellers and just a quick manual stamp at the desk on the way into baggage reclaim.
Security on the way out took maybe 15 minutes. I’d also been to brunch in Venice Beach beforehand so my mind wasn’t really with it! I would have remembered any major problems though.
I was flying NZ so the timings won’t be the same as BA and can’t comment on the lounges.