We don't have a car and can't afford taxis everywhere, so I became very proficient at traveling on the tube, both with my two kids (2 years apart) in a Phil and Ted's and with a stroller plus buggy board combination later on.
It's perfectly possible, you just get used to it after the first couple of attempts. It is, as you've guessed, safest to take the Phil and Ted's on backwards, but as long as you hold it and put all the weight on the back wheel it's fine, the escalators go slowly enough that it's not really a problem (although nerve-wracking at first!). The buggy board thing was OK for us (although I suppose depends on the make of buggy and board), you just have to make sure you step a little further back than usual to account for the board.
The worst thing was the stairs! Although as
sal says there's usually someone to help so it's not too bad. Earlsfield and Clapham Junction stations were difficult but they've both installed lifts now so I imagine they're a lot easier now