If your bags are tagged all the way to EDI, you'll sort out this hypothetical problem with BA, not MH. The final carrier is the one responsible for sending your bag to whatever address you indicate on the report, even if you travelled on a separate ticket. They're also the ones to deal with any expense claims, if it comes to that.

MSC has absolutely nothing to do with this as it only concerns baggage allowance, and even then, only if you're booked on a single ticket.