Yes LHR is a nightmare but not with CX. I go through LHR 6 times a year and I've always seen my bags on the carousel after I pass immigration and I'm not even a UK/EU passport holder. I don't even use fast track since the non-EU line is even shorter.

I've been on BA26 in J once. I waited for 45mins until most bags came out, and they weren't even in F/J/Y priority. Ok this might not be much, but this certainly isn't impressive at all given that LHR is their base and it is the first flight to touchdown in the morning!!!