Weigela and Spirea are hidden in my garden by taller plants. Can I move them? Or would it be better to layer branches and transplant those?

My mom dug up a large spirea last year and cut it up into smaller sections with each section having some root. I took a section and it bloomed beautifully this year. I believe it is Bridal Wreath Spirea, but really am not sure.

I have weigela also, but have never tried to dig them up.

I wouldn't dig them up until late summer/early fall when the weather is cooler. If you have enough space around them to dig and they're not huge, old plants then I would move them. You could always get some insurance by taking cuttings (Weigela are easy to root) or layering before you dig them up.

If I layered them now, do you think I would have a new start by spring? How long doe it take to grow?

I've only done layering "accidentally". Basically, it's weighting a branch down to ground contact with a rock or landscape pins. Not sure how long the process takes for rooting. I'm sure there are folks who know a lot more than me about it. Cuttings and layering will produce much smaller plants that will take a few years to catch up to the parent plant.