50 minutes is easily do-able, if you don't have to clear immigration and customs. Or if you have to clear immigration and you're an EU citizen and you don't arrive at a busy time. Or if you have to clear immigration and you're a non-EU citizen and you've been given a fast track pass and it's not too busy.
Some more information please, like from where to where, and EU/non-EU citizen. Is it a weekend ?

If it's all on the same ticket (ie ticketed as a legal connection) then BA will put you on the next flight as you'll be 'protected'. If it's two separate tickets, then it's booked at your own risk, and you will have to ask nicely - status would help.

hope that helps