E.O.
We're a military family, and I have lived in England for a total of 5 years. We didn't live in London, but went there frequently on day trips. I have twin daughters, so a double stroller is a must.
I have a Graco Duo Glide, an the Jeep double stroller. The Jeep stroller is easier to move around, but is quite wide and difficult to get through doors. It also has no storage. The Duo Glide might be kind of long and heavy, but it has plenty of storage, and it'll fit through the doors in the tube. But, I'll forewarn you now. You are not going to have an easy time manuevering a stroller through the train stations in England. I cannot recall seeing a lift at all of the stations. And often changing from one line to the next requires a significant change in levels, which makes for some gigantic escalators.
As for the rain, get a rain cover here if you don't already have one. They make one for the duo glide for less than $5. You'll need it. Things are much more expensive there, so get it here if you can.
E.