Edinburgh is very good around that time of the year.
But it does be packed, they have there own new years celebrations called Hogmannay (sp). I went last year, and thought it was brilliant.
Went to Rome about five years ago for New Years. It is a really busy time of year so finding a decent hotel wasn't easy. The weather wasn't great - bit warmer than here but also very wet. Most Romans head towards Plazza del Popolo for midnight. It was a bit scary as there are so many people but there was no trouble. There is also a very good atmosphere around the Spanish steps at midnight. The Italians like to party well into the night on New Year's Eve. Can be difficult to get a restaurant that night too. It was good but not something I would do again or really recommend.