Recently been to Edinburgh which we loved - so many great museums, restaurants and beautiful buildings and views. Paris is my favourite city in the world but weather wise I have been in January and it was -5. Also done Amsterdam in winter - once unbelievably cold and once fairly mild. So much depends on the sort of winter we are about to have. One place you didn't mention on the Eurostar is Brussels which is worth a look. Bruges is lovely for a weekend but less to do than the other cities mentioned. If you decide on Paris do send message if you want endless list of things to do, also for Amsterdam. Good luck with your decision.
I love Edinburgh I live betwwen Edinburgh and Glasgow but mostly choose Edinburgh if dh is driving as It always makes me feel like a tourist.The sky line as you walk along Princes street is just amazing
Well, Paris was rather glamorous when we had our honeymoon there, one November. Even the foul weather was bearable. But then, we were young and in love, and living in Edinburgh.
If you want to be talked into seeing Edinburgh, on the plus side there are loads of galleries and museums, and far fewer tourists to trip over in the off-season. The weather can be bracing, but bracing can be good if you're walking on the beach at Portobello. (Take eye protection, it's sandy and it bloody stings...) The Botanical Gardens are lovely all-year round, and there's good shopping if you're interested in same.
Glasgow at end of Jan for Burns night could be fun though. (Biased as Glaswegian.)
Glasgow and Edinburgh are close enough to each other that you could do both (1hr on train.) I can also recommend an AMAZING hotel in Glasgow that I use every excuse to stay in because its luffly and so bloody cheap for what it is.