Uber in Nuremberg and Rothenburg on der Tauber

Probably not. Where Uber is available in Germany it is just a taxi hailing app. So you will be riding a normal taxi, just ordered via Uber.
And will you need Taxis? Christmas markets are typically pedestrian affairs. You will mostly just walk. And in other places take local public transit.
And if you need a taxi, just go to the taxi rank at the station to take one, or if setting out from your hotel ask the hotel to order you one.

You don't mention where your hotels are so it's hard to give advice, but don't assume that driving is the best way to get there. And if it should be, forget about Uber and just stick to the regular taxis, that is in general the best option in Germany in my opinion.

Do the taxis in Nuremberg and Rothenburg accept credit cards? We will be traveling entirely by train and our first stay will be in Nuremberg. Plan to use a Bancomat/ATM at a BW Bank near our Airbnb. Not sure I will find one at Frankfurt Airport before taking train to Nuremberg. I avoid Travelex when I can and always look for a bank affiliated ATM.

In Rothenburg check with youraccommodation to see if they can pick you up at the station. The 2 apartments I have stayed at do. If not, there are taxis waiting at the station.

What about acceptance of credit cards by taxis in Nuremberg and Rothenburg? Are they accepted in all taxis?

At Frankfurt airport there are Deutsche Bank ATMs all over the place. Use those. Any other ATMs you see there are not advisable as you will get the same poor rates as the money exchanges. Always use euro as the conversion currency, do not fall for the $ dynamic conversion that is everyplace now that takes debit or credit cards. Your bank will do the conversion.

I would think that all taxis in Germany will take credit cards, especially in the cities. They are used to tourists and business people. I always ask first, just to be sure.

Forget Uber in Germany because sometimes they do not call a taxi but also illegally operating chauffeur drivers. Uber knows this and continues illegal operations which can end up badly for travelers.

Most taxis in Germany accept credit cards. Ask the driver before start the taxi ride or tell taxi call that you want to pay by credit card.

Tip: every taxi has a 4-digit license number in the rear window. Make a note of it or ask for a receipt (Quittung) where the number is printed on or stanced in. This way you have an identification in case of claims.

MarkK and/or others may have insight on FreeNow, but I've used the FreeNow app with success throughout Germany and Austria to call taxis. It has an interface that is very similar to Uber. It's based in Hamburg, so has good penetration in German markets (and others like the UK and Italy). Of note, though, Rothenburg is not on the list of cities served...

https://www.free-now.com/uk/cities/

Rothenburg is small. If you're hotel is inside the walls you should be able to walk there from the train station. In Nuremberg I'd also take a look at where you're staying vs. the U & S bahn stops. The problem is that with all the pedestrian areas and one way streets your cab ride could easy be 2-3 times longer than the walk. Your other three cities are much different; they are large, but if you take a look at where you're staying, and even better if you plan, you can find a place close to the public transport.

Uber is not going to save you any money. And there are always taxis outside any train station.

We are staying 15-16 m. walk from train station at Hotel Gerberhaus. Too far of a walk in early December with luggage. Will have to take a taxi.