I have all the materials to build a new pergola-style patio roof to replace my old 1940s aluminum forehead slicer. Every year, some House Finches build a nest under the current one, and with the early spring, they already have a nest and eggs.
My original thought was this:
- build new roof framing in the yard
- get a small crew to assist
- set nest aside
- demo old roof
- install new roof
- replace nest in the same spot under the new roof.
In that plan, it would likely only be an hour or so that they were disturbed. Now with stay-at-home orders, I can't do step 2 so I'll be working alone. Destroying the nest or causing harm to the eggs is not an option. Tree hugger.
How smart are they? Can I set it in another sheltered spot? Not many options for that, but there would be the top of my car tire under the wheel well about 6' away.
Ex-biologist here. Can you wait until the young fledge? That's your safest bet to not cause any harm.
I can, but would prefer to get moving on the project. She just laid a few days ago, which means 2 weeks to hatch and then 2+ weeks to fledge.
I'm just being selfish. I need quarantine projects 
I just work on cats and dogs, so I don't have any special knowledge to add. I just want to emphasize how impressed I am with your concern for these animals.
We've never met in person, but I think I can still say that I'm proud to know you.
Thanks, man. They've been nice enough to offer some good wildlife entertainment over the years. I owe them one.
I actually engineered the new roof to have a spot for them to build next year.
So sweet! I love nesting birds. We had a duck make a nest in our front yard under an arborvitae last year and I was heartbroken when a storm scared her away. I don't have any advice on this situation but I hope it works out for you and the finches.
They "should" continue to care for/sit on the eggs as long as the nest is someplace they feel safe. It's FAR safer to do it while there are still eggs and not fledgelings. Not sure about this particular group of finches, but most birds will attempt a second nest if the first is disturbed early in the cycle. So do it quick. They're more likely to come back to eggs and if they don't then they're more likely to lay again this year. What you don't want is fledgelings that you have to take care of because the parent's don't come back. There are some birds that are easier than others to hand raise but in general the smaller they are the more difficult it is. I've raised crows and cranes and all matter of waterfowl, but I've never kept a songbird size chick alive.