I've never had delayed or lost luggage with CX so I don't know what their policy is, or the legal position (as governed by their Conditions of Carriage and applicable HKG law). I do know BA policy and law quite well in this regard. Unfortunately the airline responsible for lost luggage is the one flying the last sector (CX in the OP's case); but if the ticket was purchased in the EU from BA (issued as a BA ticket number, and not CX), then the BA and EU luggage compensation does apply! So it all depends on whether this was a BA or CX ticket.

For BA, any delay of over 24 hours in baggage delivery is entitled to a set amount. The amount varies by class and route (short/longhaul). I forget the exact amounts, but the range is GBP 30 for shorthaul Y to GBP 300 for longhaul F. Receipts are required for all items, but up to this amount is paid without any fuss by BA (at least in the EU). If BA had been the carrier flying the last leg then this is the compensation that would have applied (with additional amounts for delays over 24 hours, though these days BA is more likely to just pay the lost luggage charge and not deliver it instead -- so getting the baggage at all is an accomplishment).

As for avoiding LHR meaning not flying CX, do remember that CX has good service to several other cities in Europe. FRA and AMS are good alternatives from CPH (in fact I would have prefered routing via FRA!), and even FCO is a possibility. There is no need to fly via LHR on CX!