Thanks for all the replies and suggestions.

I am flying CW so don't feel too sorry for me.

A couple of things occur to me. Why do BA schedule their flights to Japan at this time? They have a slightly later one but it's not much better? They usually schedule their flights more sympathetically - ie daytime flights to US and overnight back. I'm sure they have a reason.

Also there's no arrivals lounge at Haneda, which would have helped.

My plan is to get up early say 4 am. Get to LHR at 6 am. Spend time in lounge and have leisurely breakfast. Get head down at about 5pm UK time. If room not ready at hotel, take the shower they have offered and just get on with it. How hard can it be?