Upvote:3
Probably. This requires what is called "interlining". As of now, Qatar Airways does allow interlining with British Airways.
However, it is really up to the check-in agent, who may be following different rules, may not know the rules, etc.
So here's what to do. When you go to check in, ask the check-in agent if they can check your bags all the way through to your final destination of Karachi. You may have to show them your ticket or at least give them your reservation identifier (six letters) for the onward portion. If they say yes, you're golden. If not, they may say "your bags can't be checked through", you will have to collect your bags in London and recheck them.