London seems a little like Washington DC, but the network is more extensive and the rules more complicated. In general if you can take a bike then you can take a Segway- some lines do not allow bikes at all others not in the rush hour. Most of our stations are deep with long escallators- clearly you are not allowed to ride on the escallators or on the platforms. It seems to depend more on the discretion of the individual stationmaster. One will let you on without any problem wile another will say you can't bring that in here....
Maybe when Segways are legal they will have to think a little more seriously about the issue- a Segway is very much less of a trip hazard than a bicycle and is smaller than a lot of the wheeled luggage you see tourists with.