You can use an US PS2 on a British TV but you'll need the following:
1) A transformer to convert the power. You plug the PS2 into the transformer and then the transformer into the wall outlet. Mine cost about £30.
2) You'll also need a video adapter. Look for one that converts NTSC (US) to PAL (UK).
Connect the cables from the PS2 to the video converter and then from the video converter to the TV. The one I bought ran about £50. Also, depending on the type of TV you have, you may also need an adapter that fits the back of the TV to accept the cables.
It sounds more complicated than it is, really -- and it's much cheaper than buying a new console and replacing the games. Also, an added benefit is that if you have US DVDs you can play them using the PS2 on a British TV.
As for new games, I don't believe you'll be able to buy games in the UK and use on the US system. You'll need to buy from the States.