Pedestrianised areas normally have hours of use so, say between 9 and 5, all motor vehicles are banned, etc but outside you can drive you car about as you like. I think cyclists can use pedestrianised areas (don't quote me) but you have to give prioity to the pedestrians. If you're really concerned check out the highway code (as most cyclists should/need to). Get clued up on the highway code and you shouldn't have any probs. Plus its will make you safer and conscientious to other road users and pedestrians.
There are too many cyclists who don't know or don't care. If you're over 12 you can be prosecuted for riding your bike on a pavement, but most cyclists disregard that. As long as you give way to the pedestrians and ride safely even if you aren't meant to be cycling there, most people won't be bothered. Police officers have their disgression. If you're safe, pleasant and polite you shouldn't have any probs, be a dick and piss people off and they will fork with you.