Extract from the BA DEP requirements page. I'm 6'3 and have had absolutely no problems so far...
"" Height between 1.57m (5'2") and 1.91m (6'3") with weight in proportion to height (height is accurately determined during the assessment process). Qualified pilots who are taller than 1.91m may submit an application but will be required to undergo a functionality check to confirm their ability to meet the requirements of the seating positions in the British Airways fleet of aircraft. ""