Most likely you can move it for a few hours. Best to move about 10 AM, and have it back in place by 3PM, when they "shift change is for the nesting pairs.
I think you correct that there must be at least one baby under Mom. In my experience, they are much more prone to fly away while sitting on eggs, but most do NOT want to leave their babies.Thank you for your responses. I am pretty certain there is at least one baby in the nest. No matter what I do the pigeon will not move more than an inch or 2 to the side and it won't stand up. How long do they sit on the babies around the clock? I hate to disturb them at this point, if the mama will even let me. Maybe I can delay the repair a week or two until the babies are a bit older. If I wasn't pregnant myself I would give them more time but I'm having my own troubles dealing with the heat!