What to replace an oil heater from 1989 with in an old witch's house?

Hmm an external chimney is probably not an option here. For me, it’s mainly about calmly devising a good plan now that I can then implement when action is needed for the heating, instead of making hasty decisions at that time.

I will now check what a gas connection would cost; a gas tank in the garden is rather unappealing to us. I have read some success stories about heat pumps in old buildings; if it can be implemented sensibly in our case, I would really like that solution. We have a long transition period here and I think that at least from mid-March to November I could manage exclusively with the heat pump, and it would pave the way for a nice wood stove in the living room, which can then be heated in winter.

It is important to us to have a reliable heating system that also reliably protects the house from freezing/mold during longer absences. Pellets seem less suitable to me; gas, oil, and electric are probably ahead here?

I will now check what a gas connection would cost

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pave the way for a nice wood stove in the living room that can then be heated in winter.

You really have to want that. My relatives do that, and the whole house always smells like a fireplace. I would neither be interested in the work nor in the smell. Even if the system is better sealed than I know it, that would be too much effort for me.

It is important to us to have a reliable heating system that also reliably protects the house from freezing/mold during longer absences. Pellets seem less suitable to me; gas, oil, and electric apparently have the edge here?

Any heating system can keep a room above freezing. Ventilation helps against mold. That is independent of the heating type. A pellet heating system also runs on its own if you are not there!

I think you say pellets but think firewood. With pellets, there is usually an automatic feed auger that fills the stove without the involvement of the residents. The disadvantage of pellets is more the space requirement (which is almost prohibitively expensive in new buildings), but that should be less of a problem for you since the oil tanks also have to be somewhere and it might be possible to convert that. However, it is definitely necessary to clarify whether you have space for 4 tons of pellets (which corresponds to 2000L of heating oil according to the rule of thumb) and whether you can also properly install the feed system and heating.

There is space, but the internet is full of people whose [Förderanlagen] fail, and I also see that in my circle of acquaintances. From my understanding, a pellet system should run as much as possible at maximum load to be efficient. In my eyes, that somewhat conflicts with the long transition period here.

As long as the old oil heating system is still running, I wouldn't change anything about it. Afterwards, I would use a heating technology that requires as little maintenance as possible and is suitable for the building; in my opinion, that would be a gas heating system operated with liquefied gas. If the gas tank in the garden is too much of an eyesore, there is also the option to place it underground. Of course, this is associated with higher costs.

Originally, liquid gas was also foreign to me, since I come from BaWü, but now that we live in Saxony, I feel like I see it in every second front garden here in the countryside. In the new development area, about half of the builders have buried a liquid gas tank in their garden. We also have liquid gas here in the transitional apartment, and eventually you no longer notice the tank; the heating runs steadily. Therefore, liquid gas could be an option. Your mother usually lives there alone and you are only there occasionally, which would also incline me more to keep the existing system running. With the old roof, the consumption was +/- 2000 liters/year, which in my opinion is quite acceptable and should now be improved. It’s also not the case that a renovated heating system is comprehensively environmentally neutral; there is some deception involved there too when considering the whole picture. Now, heating behavior, detailed situation, etc. are not known, but if you carry out some insulation measures that you would have done anyway, the consumption could already be reduced enough to be fine. One really should not believe that all newer houses are so wonderfully energy efficient, and if they are, they tend to be lived in in an energy-damaging way by users through high room temperatures, high electricity consumption due to excessive lighting or technology, etc., all things your mother probably will not have. Every type of heating has advantages and disadvantages or particularities in handling and function.

I have read some success stories about heat pumps in old buildings,

There certainly are such stories, but these exist for all other options too. Ultimately, one’s own preference for a system also decides; there is no generally better one. It is also true that your mother might need something different than you, and fortunately, you can still observe the development a little longer.

We have a long transition period here, and I think that at least from mid-March to November, I could manage exclusively with the heat pump, and it would pave the way for a nice wood stove in the living room that could then heat up in winter.

Precisely with this “scenario,” I would look very closely to see if it suits me, also since you presumably don’t have underfloor heating there. But there are people here or in a specialized forum who can tell you exactly about that.

It is important to us to have a reliable heating system that reliably protects the house from freezing/mold even during extended absences.

Exactly THAT you should consider closely, and THAT is where I would have my doubts regarding a heat pump. That would rather speak for electricity + photovoltaics again, because this system probably has the lowest vulnerability to damage. Like my predecessor, I would simultaneously vote to keep the system as it is and then use gas or electricity. But you still have time. :D