Originally Posted by
ajeleonard
1st option using the arrivals lounge is going to be a lot less hassle
OP doesn't mention flying business or having OWE status, so may not have lounge access. However even without, I'd agree that going landside straight away is better.
I'm also not sure why (assuming no lounge access) OP wants to go somewhere to get some rest before heading into the city. I think it would be better just to get onto the Tube - which provides an hour long rest - and then find somewhere nice for breakfast in town.
12.30 isn't too late to get on the Tube. You have to calculate timings backwards. So if your flight is at 15.10, you need to be scanned at the entrance to security at T5 by 14.35. Giving yourself a good half an hour leeway to collect bags from left luggage means you should aim to arrive at T5 no later than 14.00. The journey is an hour from Kings X, 50mins from Green Park, 45 mins from South Ken, so aiming to get on the Tube by 12.30 is fine.
I'm not sure what the exact impact of Ride London will be, but I would generally try to avoid the route pinch points lest you get stuck in crowds - check TfL for advice on those.