You need hydrogen to produce synthetic methane, and plants to produce it. The larges source of H2 globally is from you guessed it, natural gas. To get if from electrolysis, you need an electrolysis plant near your synthetic methane plant and both need to be near water because it's difficult and dangerous to store H2. All of that would take more than months. You could probably replace a large number of boilers in 3 months. Not enough to solve the problem, but enough to make a dent. Electric boilers, while less efficient than heat pumps, only cost around $700 each and are basically a 1 for 1 replacement for a gas boiler.

They sure could, and may end up doing that but I'd bet it's going to be politically unpalatable.

Both retrofitting and coal gasification could probably be done in parallel.

Which is still infinitely better then never.

Which is still, to be fair, probably the fastest conceivable solution. And I am far from convinced about nuclear.

Germany is in a unique position here, having recently shut down over a dozen nuclear plants. Three months is probably not enough to refuel, restaff and cut through the red tape, but it doesn't have to be decades.

You could take the serially produced reactors that are used in ships. If you pay enough you should get the next batch. If you push it like China pushed its emergency Covid hospital, it should be a matter of months to have some reactors going.