Well,I stand to be corrected,but when I worked for BA at LGW (and this applied to all BA stns) anything left ON THE AIRCRAFT was BA's responsibility, only items left in the terminal building were BAAs responsibility....unless things have changed,the coat should have been handed to BAs own lost property...and in my day we would indeed have checked the pax booking to see if they had another sector, and commented where the coat was being held.I wouldn't let this lie if I were you..it's gotta be somewhere..