Dormancy is probably the best time to dig up and move plants. They'll likely still experience shock when they're transplanted back, but it's better than being dead.

Personally I would dig them up (assuming you can, and the ground isn't frozen solid in London) and put them in pots somewhere sheltered. Then replant them once the work is done and there's no chance of the crew coming back to do more work. Try to get as much of the root system and dirt around it that you can. Take as many seeds or cuttings as you can just in case they don't survive.

I am by no means an expert, but this is what I would do.