I’ve been to all 3 places, and they were all equally good (and unique).

It’s a tough call for New Year’s. Prague sounds like it would be great, but I’m afraid it will be too crowded. The public square in the castle district is not so large compared to how big the crowd might be. I was in Budapest for a big national holiday with fireworks and everything, and it all took place along the river, so there was room. Budapest was also much hotter during the summer than those other places, so it might not be as cold.

I wouldn’t let the weather keep you from going. I’ve travelled in Europe for New Year’s before. Just bring lots of layers of clothing (like thermal underwear, hooded sweatshirt, jacket, thick socks) and maybe keep a thermos of hot coffee with you and you’ll be fine.