Thanks for confirming that this would be my only option. I did this once back a couple years ago, and it was ok. Having gotten up so early and spent so long awake on the plane, I was tired enough to go to sleep when I got to London, so it all worked very nicely. But I do generally find Heathrow a pain and getting from there into the city and back is also not so efficient. I stayed with a friend last time so no additional hotel costs (and a chance to see how my friend lived in London--he has moved in the meantime to another place in London).
Taking the train instead of a connecting flight is also an interesting option. I did that from CDG once, which is only a four hour train trip, which drops me a ten-minute walk from my home, rather than at the Frankfurt airport. I think the Eurostar doesn't have as good a special fare deal as the Paris route, but it is worth looking into, particularly if there is a decent connection (I could sleep better on a night train than on an economy flight, and I'd rather not use miles for upgrades when I am travelling solo).