Do you mean touring or riding your bikes around the city?
If you mean touring then I'd say it's not a good idea. Cold, wet, possibly snow, short days, pretty miserable.
Around town you could most likely ride unless there's snow/slush/ice. Why bother though - Edinburgh city centre is so compact that it's all walkable
+1. And London roads are so congested that for city sightseeing, it's pretty undesirable. There are Santander self-rentals all over, though, so if you want to cover some particular ground, you can do that.
Now, riding south of London in Kent presents some very nice opportunities on hilly roads and touring destinations to castles like Hever. Whether you want to undertake such trips in Dec/Jan requires serious consideration, though, as many tourist destinations are not open, and the weather poor.