The Canaries should be over 20 degrees in January, but you may be unlucky and get some cooler weather.
Otherwise, from Newcastle I can only think of Egypt. I have been put off because of the security requirements, but also know many people who go every year and love it?!
Ideally I would go to the Caribbean - but I expect you would need to fly from Manchester.
Problem is that for many winter destinations, January is their high season, so it is expensive - and from prior experience, flights to the Canaries sell out quickly for that month.
We usually go on a long haul holiday in the winter, as that is when we are not milking our cows - and it is difficult to find somewhere affordable (we want to go to Costa Rica, but you are looking at several thousand pounds!). We went to our beloved Spain last January (flew out on 2nd January, just as the worst of the snow hit Wales - got the last flight out from Cardiff for a few days), and it was 18 degrees most days. I was in an outside hot tub in my bikini whilst the UK was struggling in the snow. Not beach weather, but still a wonderful break away and so much quieter.
Another option could be Dubai - there is lots of choice there, so you could get a good deal.
Ali xx