Lean mean,
Don't worry about it. I'm 6ft 5 and have checked with everyone I can think of about it and no one other than BA imply this restriction.
It's interesting that for DEP at BA the restriction was removed.
I know of people taller than me that are flying so it's no biggie (no pun intended, honest). Apparently OATs had a guy who was 6ft 7 do thier course.