For my last 3 LAX -LHR flights, with the last one been November last year, check in / bag drop opened approx 3hrs before departure of the 1st flight of the day.
There are now SSCI machines for BA flights at TBIT. Although when i used them back in November it would print out my LAX - LHR boarding pass but not my LHR - MAN boarding pass. Despite the 2 flights been on the same PNR.