Can you put it back quickly before the mom forgets about it and builds another nest?
I would not move it unless I could move it a few feet at a time maintaining about the same environment (amount of shade) each time. Wait a few days and observe that mom has settled onto the nest again between each move.
The average homeowner who lost his house in the Oklahoma tornadoes should move for good and not rebuild. Too much complexity watchdogging the contractor. Too much a chance to be defrauded.