Are you doing the best thing by taking the overnight flight? Take the service the next morning, enjoy the J cabin to London then get to your hotel around 5pm. The hotel room will be available so have a shower & head out for dinner. Yes you'll be tired, but you only need to last a few hours until rolling into a proper bed.

The overnight flight might get you an extra full day in London, but if you spend a few hours in the arrivals lounge, the overall benefit is going to be limited.

What I would add is that when I did take the CX flight into Gatwick, the airport was VERY quiet on arrival which won't be the case at Heathrow at 6am.