It will be cold in NY, yes (although this year has been exceptionally mild). Your DC are the right age to enjoy NY at that time though, and you/they will have plenty of time to look into what they want to do, where they want to go, book tickets etc. Assuming you’re coming from the UK, 4 nights could be brutally short in terms of jet lag. You’d just have to power through and rest when you get back.
£4K for 4 ppl, flights and accommodation is ambitious. Sorry. Flights are around the $750/800 mark now for non-peak travel days. Accommodation will also be super expensive over the NY and you really don’t want to mar your experience by staying in a motel or hotel with bedbugs or inescapable noise etc. If you insist on traveling over that time, fly back on Jan 1, or fly out on Dec 25. It might help with costs. Then, of course, spending money. Boys that age with large appetites….stick to fast food chains (they’re not all awful, and it’s only 4 days, plenty of salad chains as well as burgers and tacos etc), local bistros etc. Portions are big, at least.
Iceland I think would be more realistic BUT you may have to spend $$$ on cold weather gear if you don’t have any. It would be miserable to shiver throughout your trip. Boots, coat, ski pants etc - if you don’t have them yet, beg steal or borrow! Also, no guarantee you’ll have a clear night and if you do, that you’ll see the lights so have a Plan B up your sleeve.
What about Svalbard, or Tromso? Northern Finland? You’ll need all the same winter gear but it’ll be cheaper. Also if you don’t see the lights you can go dog sledding, reindeer racing, snow-monolingual etc. But again, £4K would be ambitious.