Technically BA's on-line check in is a doddle but over the last few months the one thing it isn't is fast. Suggest you go on line 24hrs before and select your seats/check-in. Once you arrive at the airport go to one of the automatic check in machines - dead easy to use and will print your boarding passes (just a word of warning to check your seat numbers at this machine vs those you checked into 24hrs previously - they have been known to change) then take your bag to the (anything other than) fast bag drop - this will probably be highly visible due to one member of staff and a massive queue Apologies to anyone who works there and disagrees but my view is your management need to get their priorities sorted).
Once through this you just need to queue for security - very hit & miss - on a weekday morning I've walked straight through and I've queued out of the door and half way to Brighton.
Please excuse the cynicism - I'm sure you'll enjoy our world leading partnership of BA & BAA !