Sort of Christmas vacation

While I can sympathize with a desire for a getaway Christmas trip, as first timers you've left your planning pretty late. You say you've never been abroad, but do you have your passports? I'm afraid that a trip to the UK over Christmas, with last minute flights and hotel bookings, do not go hand in hand with "as cheap as possible ".

Check Google Flights for flight options into the larger UK airports like Heathrow, Manchester and even Edinburgh.

Check booking.com for accommodations in whatever city you decide on.

There are literally hundreds of places where you could stay, but cities are usually more condusive to a short winter stay than rural towns They will also have an abundance of indoor attractions if the weather is inclement. And there will be a greater variety of dining options. I doubt you will find much in the way of "extreme" dining anywhere in the UK, even though the larger cities will have an immense variety of cuisines from all over the world, if you choose to go that way. You can also find all the usual American fast food joints.

Be aware that just about everything completely shuts down from Christmas Eve to Boxing day, so don't plan any travel then if at all possible. And reserve your meals in advance for those days. Many restaurants will be closed.

Seeing the Northern Lights is totally weather dependant and needs the right atmospheric conditions. You would need to be away from the city to see them even if they appear. You can Google the odds of there being a white Christmas anywhere in the UK. They arent great, but more likely in the north of Scotland than the south of England or Wales.

Check with itur and see if they have any last minute package deals. I'd aim for London, Frankfurt, Amsterdam, or Copenhagen, and just plan to stay in that city for the most part. All of those will have Christmas decorations, etc.

Snow is not a given, unless you add someplace like Zurich to the list. But those places that can guarantee snow will be in high season for skiing, and not inexpensive.

I was going to say there were a lot of places that present the possibility of meeting your objectives, but each comes up a bit short one way or the other.

We would like someplace snowy and cold, and decorated to the nines for
Christmas.

Most of Europe will have markets or lights. Google Christmas Markets in Europe and see the options. My Budapest scores on that one.

Somewhere the food is not too extreme, as my son is fairly picky. If
he can get a burger and fries he will be happy.

The world now eats American. No problem anywhere these days.
Again, my Budapest hits on this one.

We would like to maybe see the northern lights,

For that you have to way far noth. I am not an expert on the best places for this. I am certain someone will chime in.

do some walking/shopping/exploring.

Again, look for cities with Christmas Markets going on. And for cities in general. European cities are a lot more pedestrian than American cities so walking is a given.

Another plug for Budapest (I am somewhat biased. Whenever I write Budapest know that there are a lot of other cities that hit the same points .... except in the last comment below)

Maybe ride a high-speed train Just somewhere we can get away as cheap
as possible and enjoy some beautiful scenery,

I am one of those that thinks the view from trains are over rated. Generally the are buffered from the surrounding countryside to keep noise down. Not much to see.

some beautiful people, and some delicious food. Does anyone have any
suggestions? Thank you

the world is full of beautiful people and good food. So there is no limit on that.

get away as cheap as possible

Much of Central and all of Eastern Europe (Budapest for example, or Prague .... BUT NOT VIENNA) will be 30%, or more, less expensive to visit than Western or Northern Europe.

Someone said it might be too late. Well, it might be too late to get good deals of flights. I like google flights. They have a matrix that will show you the cheapest combination of departure and return dates for any destination you put in. That will help a lot this late in the planning.

As for hotels and other tickets, yes, your choices will be reduced this late, but its far from impossible to find good deals. Just takes a little extra effort. If you know of anyone that travels often, get their help. You also might stick to locations where you know someone can help with a lot of the details, as its late to spending a lot of time researchng.

A final suggestion, since price came into the converstaion, and I might be reading too much into it, but ..... The UK is expensive, A good winter tour is Prague, Vienna and Budapest. All three have great winter markets and the chance of snow exists. From Vienna you could take a side trip to higher country to see some snow. These three seem to work for you because you get to ride the train between cities and because, even while Vienna is one of the more expensive destination is Europe, Prague and Budapest are among the least expensve and will help balance the overall costs. You can do this in 14 days. Less time? Skip one city. More time? There are some good side trips out of each city.

In the UK there is pretty much nowhere where you can expect it to be snowy or even cold (depending on your definition of 'cold').

The only place where there should be snow is in the Highlands of Scotland- the Cairngorms, the Lecht that kind of thing. But with climate change even there it is unlikely to be "deep and crisp and even" and unlikely to be such that you can ski on it.
I've not been to Inverness at Christmas but I would be surprised if it was "decorated to the nines" or indeed any of the towns in the Cairngorms. Nicely decorated, but not to the nines.

I'm not going into the food trap (we had a pretty bad thread on that a few weeks ago) but pretty much anywhere you are going to find a burger joint type of place, or other global type food options. I will say that going to a new place part of the travelling experience IMO should be trying the local food, and will leave it at that.

As for the Northern Lights they have been appearing across the UK this year. But in a normal year Orkney and Shetland are the best places for the Lights. However they do not get snow, or suffer from the extremes of cold which the likes of the Cairngorms do. And even that far north you will get burger and fries- often with a local twist to the burger as an option.

Edinburgh is a great place for Christmas, where public transport operates pretty much as normal 365 days a year. You could ride the High Speed train from London (relatively speaking- the UK does not have High Speed in European or Global Terms- think the Acela if the whole route from DC to Boston was rated for high speed running). It is a place where Christmas decorations will be in full swing. You can then do day trips to the Highlands from Edinburgh.
It is off season so you should get pretty decent rates- especially if willing to stay in a budget hotel chain like Premier Inn (for instance down at Leith).
The perceived problems of length of train ride, and issues caused by engineering work are often heavily over-stated on this forum.

But in the UK Edinburgh is probably the best place to optimise your particular specifications.

The old standby of London would meet the "decorated to the nines" and of course the food , but not most of the other wishes- especially snow and cold.

They didnt say for how long or where they were coming from

And since they have never been to Europe the UK might be a bit more comfortalbe as the language there is so similar to English. Again, because they havent traveled a lot, maybe London as a base, near a train station where they can do some day trips or an overnight someplace and still have an area they can return to that is well suited for a longer stay .... and has a KFC or a McD.

And while the UK is going to be a lot more expensive than Eastern Europe, the flight over might be a lot less expensive and make it a draw.

But they have so little time and so much to learn they really need a trip planner or someone on here that will take the time to tell them exactly what to do, where and how to do it. Or they could gamble a bit, buy the plane tickets now, the do the rest on the fly.

If i were them I would run costs on the UK (with tickets) and on Prague + Budapest (with tickets).

What a lovely idea!

As mentioned, flights will be your biggest cost. So maybe take a look on Google Flights by using your home airport (or best international airport near you) and for the “to”, type in Europe. It may give you ideas. There are any number of cities that would hit most of your wishlist. Sometimes airlines run Black Friday sales.

Once you have flights, find lodging. If you are there over actual Christmas and Christmas Eve, you might prefer to find an apartment as opposed to a hotel (so you have back up cooking options if restaurants are closed). An apartment will generally give you more space for your money, as well.

It’s not too late at all! If you have flights and lodging, you can add the rest last minute!

Ideas: London, Edinburgh, Stockholm, Copenhagen, Helsinki, Tallin, Berlin, Frankfurt, Munich, Vienna, Budapest, Paris. None have 100% but most of them have most.

You should probably think about what you're really looking for. If you're going to the UK, it will most likely not be snowy and cold since the islands have a mild maritime climate. And while the northern lights can be seen in the UK when there is a strong event, you need to go much further north if you want a decent chance of seeing them.