I was wondering if I could buy a pre-paid cell phone - with a London number
There is no such thing as a cellphone with a London number. Unlike the U.S. where cellphones are just a regular prefix within whichever is their home area code, the British network places cellphones in their own separate "area" codes. These pseudo-area codes and the following digits identify the particular network involved, but not the location within the country.
So when you see 020 7222-xxxx you know it's in London, and in fact if you know the prefixes you can tell that it's Westminster. When you see the cellphone 0776 706-xxxx though, you can identify it as being Vodafone, but that's it. It could belong to somebody based anywhere in the country.
The practical implications for use are:
1. When calling
to a cellphone, whether from a landline or another cellphone, you must always dial the full "area" code plus number, e.g. 0776 706-xxxx. This is so even if you are calling
from a cellphone with the same "area" code.
2. When calling a landline
from a cellphone, you must always dial the full area code and number, e.g. to call 7222-1234 in London you would need to dial 020 7222-1234, even if you are in London yourself.
In case you're not aware of this, cellphones in the U.K. (and indeed in much of the rest of the world outside of North America) work on the caller-pays principle, i.e. you are not charged (or deducted minutes) for incoming calls to your cellphone, but the caller pays more than he would to call a landline.