Uyuni flats and Licancabur climb during Christmas

you are going to find it difficult to find anyone to take you to do this. Most companies are stopping tours from 23 dec to Jan 6th to allow their drivers to go and get drunk at home instead of whilst driving tourists across a desert. Plus the kind of tour your talking about is usually a private group so if you are just two people then your looking at something pretty expensive (you will pay the cost of about 6 people)
A handful of agencies will operate over xmas and new year but you will definitely increase the risk of your driver getting drunk, even more so if your off climbing a mountain for a couple of days - he is likely to be in a semi coma state when you return.
Im not saying its impossible but im saying its unlikely and risky for you to sort this the way you are trying to.

oh, this is very disappointing. Out plan was to get to La Paz on 16th Dec, and first go to Lake Titicaca for a week, and then do the climb. Based on your relies looks like we should first go to the salt flats and to the lake later.

Unless you live at altitude or have already acclimatized elsewhere when you get to La Paz, I wouldn't advise you to climb to the top of a nearly 6000 MASL volcano only 3 or 4 days after arriving at altitude. That's not just very hard work, but also seriously dangerous.
So unless you've already acclimatized, you best start thinking about a plan C, as your plan B is no good either.

Normally to climb the Licancabur you have to take a local guide from the area which are based at laguna Verde. Around December most of they head back home to there village or Uyuni, meaning if there's no local guide around you can just head up by your self. Drivers can take you up to the end of the road from where the climb starts. You then need to make a time to meet the driver on the way back down. There can be some snow in December, meaning you might need crampons.

The guides charge around $60 for the day.