> Would they have been able to shutdown coal/gas faster if the nukes were kept running?
Probably, but it's not just the politicians and the population who don't want nuclear power, it's also the operators. They're not interested anymore as they can't plan ahead. If the current government decided that the nuclear power plants should run longer they would all need to get modernised first, meaning the operators would have to invest and who says the next government won't shut down them down again. Nobody is going to invest in building a new nuclear power plant that may or may not enter operation in 15-20 years.
> Could Germany reduce its dependency on Russian gas with more nuclear?
Short answer: No. Germany mostly relies on gas for heating, not for electricity.