> If you want to build more nuclear plants at the same time you also need more workers
So hire more workers. Most "nuclear plant construction work" isn't some foreign species of construction work. It's pouring concrete and laying pipe. The main part that isn't is the reactor vessel itself, but that's the part that can be made on an assembly line and shipped in.
> more specialized factories
Building a factory takes what, a year? Then it's mass production.
> a proven design
This is literally the main reason construction takes as long as it does already. If it isn't accomplishing that then throw that process away and build them all in three years with plenty of time left over to correct any issues before bringing them online.
> If you assign a worker to Hinkley C then he is going to work at Hinkley C until it is finished. You can't just send him to a different plant. That would delay Hinkley C.
Fortunately there is not only a single worker in the entire country, and arbitrarily many can be trained on a scale not measured in decades.