Christmas and New Years in Europe Itinerary Help

Christmas and New Years can make museum / attraction opening times a
bit wonky

That's a bit of an understatement. I've never been to Austria over Christmas, but in many European countries it's not uncommon that more or less everything is closed over Christmas.

You might also want to reconsider the day trip to Salzburg, there is a high risk that the trains will be extremely crowded on the 23rd.

First a direct answer:

December 21: London -> Vienna (Flight)
December 22: Vienna
December 23: Vienna -> Salzburg -> Vienna (Train)
December 24: Vienna If the Vienna experts tell you that restaurants will be closed for Christmas, then rearrange your plans to be in Budapest, where the best will be open.
December 25: Vienna
December 26: Vienna -> Bratislava? -> Budapest (Train) Get an early train and skip Bratislava, the time is better spent in Budapest.
December 27: Budapest
December 28: Budapest
December 29: Budapest -> Prague (Train or Flight?) If there is a non-stop flight on the 29th, then absolutely a flight. Faster and I think less fatiguing.
December 30: Prague
December 31: Prague
January 1: Prague -> Paris (Flight)
January 2: Paris
January 3: Paris
January 4: Paris
January 5: Paris -> London (Train)

What I would do. Actually I would move the date up 3 days to start on the 18th to get to both the Vienna and the Budapest Christmas markets in full swing, but using your dates:

December 21: London -> Vienna (Flight)
December 22: Vienna
December 23: Vienna
December 24: Vienna
December 25: Vienna -> Budapest (Train) Good use of Christmas morning and you were short a day in Budapest.
December 26: Budapest (the Christmas markets sans food, remain open)
December 27: Budapest
December 28: Budapest
December 29: Budapest -> Prague (Flight)
December 30: Prague
December 31: Prague
January 1: Prague -> Paris (Flight)
January 2: Paris
January 3: Paris
January 4: Paris
January 5: Paris -> London (Train)

As for crowded trains, buy tickets and seat reservations and it doesnt matter if they are crowded.

You have sort of shoehorned Prague into the schedule. If that is where you want to be for New Year, then great. Personally I would rather be in Paris where New Year is likely to be a little more "adult". My fear in Prague has something to do with it being so popular with the European trust-fund drunk brats. If you do skip Prague, there are direct flights to Paris from Budapest wich are very cheap. Then you could add a day to Budapest and a day to Paris.

Budapest over the holidays
http://budapestchristmas.com/
https://www.budapestbylocals.com/christmas-markets-in-budapest/
https://edition.cnn.com/travel/article/budapest-winter-things-to-do/index.html
https://www.budapest.org/en/events-budapest/christmas-markets-budapest/
https://adventbazilika.hu/en
http://www.budapest-discovery-guide.com/christmas-in-budapest.html
https://budapestlocal.com/9-reasons-visit-budapest-winter/
https://funzine.hu/en/2023/01/03/goodapest-en-2/winter-in-budapest-5-fun-things-to-do-outside/
https://welovebudapest.com/en/toplist/10-best-things-to-do-in-budapest-this-winter

Just a few comments -- While Salzburg is well worth a visit, I think it is just too far from Vienna for a good daytrip. Bratislava is an easy daytrip from Vienna (just one hour each way). My wife and I enjoyed doing that earlier this year, however, I should note that what we liked about it was walking up to the castle, the views, and exploring the old town. That was in pleasant spring weather and I am not sure I would recommend it nearly as much in late December; there isn't much there in terms of interesting museums or other indoor attractions. Of your four major cities, Vienna and Paris are loaded with the latter, the other two not quite so much.
I have been in Paris for New Year's Eve and didn't like it. It was hard to find a restaurant that wasn't booked up, and virtually all of them offered only a very expensive fixed price menu. The streets were overrun with wild youths throwing fireworks everywhere. This was some years ago, but I would suspect neither of these things has changed much. Can't say Prague would be any better though, haven't been there then.
In general though I think that while your plan to visit four of Europe's major tourist cities (plus maybe two other cities) with your family over the holiday period is an ambitious one, you have more than two weeks to do it, so it's not unreasonable.

@Badger - Thanks for the advice! I'm aware of Christmas closures, but I've heard that some things are open around Christmas. Will look into that further once I've finalized the dates. Also, I just realized how far Salzburg is from Vienna--definitely reconsidering the day trip.

@Mister E - Thanks for the very detailed feedback and helpful changes to my itinerary. Unfortunately, my parents will not budge on arriving in London on the 19th and leaving for Vienna on the 21st. My dad has relatives in London, and he wants to spend a day with them (the 20th of December), so there's not much I can do about that. As for Prague being shoe-horned in, it's an unfortunate consequence of having the balance where everyone wants to go: I want to go to Vienna, Dad wants to go to Prague, Mom wants to go to Budapest, and Mom and Sister want to go to Paris (which I'll have been to already).

@Slate - Thanks the tips! May also reconsider Bratislava, though I will note that my parents love cold weather and are pretty unfazed by it. About Paris on New Year's, funny how your advice was the polar opposite of the Mister E's one. I'll have to look into those more. Glad you find the basic overview to be ambitious but reasonable, which is what I was aiming for.

Have you booked places to stay over the holidays? You should make those arrangements soon.

About jet lag, 2 days isn't necessarily enough for some people, and it gets worse with age. Look at the 'Tried Timeshifter App' thread by Pam, lots of good info for your family.https://community.ricksteves.com/travel-forum/general-europe/jet-lag-i-tried-the-timeshifter-app I think NYE is a relative issue, Paris might be mayhem but maybe Prague is even more so?! My personal solution is to find a pizza place that opens at 5PM, have an early dinner & get back to hotel early enough to watch a good movie in the room. You may decide to split up sometimes for part of a day, (good idea anyway), so you can go out with the revelers & others can get back to hotel.

I'm not sure about the closing dates for Christmas markets in Vienna, but check, that might be a fun thing to do. Christmas Eve - Yes, most but not all museums, stores & many restaurants will be closed from early afternoon & many don't open again until December 27. December 26 is a holidays in some places, but sounds like Budapest will have a lot open. AND churches are open, either for visitors or services, which can be gorgeous at the holidays! 4 hotels for 15 days sounds doable, though speedy. Hopefully your family knows how to pack light?? Glad they revel in bad weather, it won't be awful but it might be windy, wet & cold. Sounds like a great trip.

At least in Budapest the Christmas markets will still be open until NYE. But not food stalls.

Let me know if I can help. I will be here.

Years ago, we spent Christmas in Budapest and then New Years in Vienna. We had dinner at a lovely restaurant with live music on Christmas Eve, and we were able take a boat cruise and visit Christmas Markets on Christmas Day. The day after, we saw the Nutcracker at the Opera House and went to the baths.

At the time, there was an entire website devoted to New Year's Eve activities in Vienna. We chose a fancy dinner with rooftop firework viewing and more food and champagne at Midnight, but the possibilities were endless. We were still able to visit all of the Christmas markets in Vienna between Christmas and New Years. Vienna is very, very busy the week leading up to New Year's Eve and we found everything was crowded - best to make some dinner reservations.

Last year, not at Christmas, as part of a bigger trip, we saw an Opera in Vienna, then moved to Bratislava for 2 nights, which we enjoyed, then took the train from Bratislava to Prague. We enjoyed seeing the countryside from the train and 1st class tickets were one of our all-time bargains.