Your checked baggage should be checked straight through to HKG (BA at MIA should not object, even if there is a separate ticket involved, because you are travelling on CX).
For any hand baggage you want to leave at LHR, all 4 terminals now have left luggage facilities, outsourced to a company called the Excess Baggage Company. Exit through customs at Terminal 3 and look straight ahead of you as you enter the public arrivals hall. You should see the Excess Baggage Company store (bright blue signs with bright yellow writing). They will want to X-ray your bage before storing it. The charge is GBP4 per 24 hours per bag and they take credit cards. They take any size of bag up to the 32 kilo limit imposed by the airport for baggage handling.
Contact details and more information here:
http://www.excess-baggage.com/airpor...l_services.php