Flying BA long haul with a 9hr layover. Depending on weather, mood, and energy, we were planning on making a trip into London since we've never been there. That being said, I was wondering if we should:
1) deal with the 6AM arrival rush and immediately go landside and then find a place to rest for a bit before heading into the city or
2) go through FCC, get some rest, then take the train out/back to T5B/C to reach immigration?

I figure the latter also requires having "ready to fly" removed (does the former too?). Should we do this as we leave the airport or once we return?

Also, our onward flight is scheduled for 15:10 or thereabouts. What would be the latest time we should be on the Tube for the ride back to LHR? Is 12:30 too late? We'll have to collect our carry-on bags from left luggage too.

If it makes any difference, this happens to be on the Sunday of the RideLondon event.