There were several studies done after the shutdown of the nuclear power plants, that showed, that the electricity prices did not increase.

Eh, it depends a lot on what exactly they analyzed. Throwing away electricity you basically already paid for is gonna cost you, there is no way that can be circumvented. It is not like we have so much wind and solar energy in the mix that nuclear could not have replaced more expensive gas, coal, oil, biomass, whatever.

Finnland has a comparable electricity price, but a much higher proportion of installed heat pumps.

Household electricity prices in Finland were a lot cheaper than in Germany up until the gas crisis, which is unrelated to my point about nuclear. Here is an example from 2020:

France is a bit weird, I think they actually heat directly with electricity a lot. I guess that's a case where electricity is TOO cheap so people use it in stupid ways. :) Too much of a good thing can turn bad as well, I guess. Would not have happened in Germany even with extending nuclear, though. The thing is, heat pumps in France would not change much about their emissions. Heating (mostly) with nuclear electricity does not emit more than heating with heat pumps.

The whole uproar about this was synthetic, newly constructed houses install to over 50% heat pumps and only 10% gas. Electric cars would be adapted more, if companies would sell small, cheap EVs as well.

This does not mean extending nuclear could not have helped (assuming it would have helped to lower prices, which I still assume here). Maybe people would be more eager to replace their gas heaters with heat pumps if electricity prices had not been going up all the time (a lot more than in basically all other EU countries) in the past 20 years, what do you think? New houses are a special case anyway, since you basically already have to design them in a way that makes heat pumps the better option.