Q.Will a Pussy Willow survive being replanted?
My Pussy Willow bush-tree fell over during a storm last week. It didn’t break, it just fell over from the ground being soft and wet. Since the bush was in a flower bed along the house, we decided to move it to a place where it could grow better. We were very careful digging it up, but when we replanted it, all the leaves wilted and are now falling off. Is there a chance that it will survive being moved?

Your plant may be stressed. Provide daily water for at least two weeks, then you can cut back. You also could cut back the branches so the shrub has less work to do. Pussy Willows are very hardy and your shrub should make a full recovery.
Ironically I had the same thing happen with one of my Pussy Willow plants this week with a very heavy rain. I did leave mine in it's current location, but a few branches were bent. I opted to trim all the branches back to about half the length.
Here are a few links for you.
https://www.gardeningknowhow.com/ornamental/shrubs/pussy-willow/growing-pussy-willows.htm
https://www.gardeningknowhow.com/ornamental/shrubs/pussy-willow/how-prune-pussy-willows.htm