Oh yes you can!

Pretty much every Tube station now has credit card- (and note-) accepting ticket machines, however small the amount. Until the chip & pin readers are installed, the only proviso is that you can only use your card once per day in a given station (unless you go to the counter).

If you need taxis that take credit cards, say that you want to use one before entering (you normally state your destination before entering in London, anyway). You will be charged a supplement.

Where you're most likely to come unstuck without cash is for small incidentals during the day - take-away food, newspapers, drinks, etc. You can find somewhere that takes cards if you try hard enough, but it may not be that easy, and they may have minimum spends.