I am travelling with Qatar @irways and British Airways on two separate tickets. On my way back from Vancouver on British Airways to London I will be travelling from Heathrow to Doha and Karachi. Will my luggage be transferred to Qatar or do I have to physically pick it up and recheck it onward to London, Doha and Karachi? Or can it be transferred to the onward flight?
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.