I had one with an egg and baby outside my window and it has disappeared over night. However, I do have 4 cats that I hope didn't get it. No, birds can't move their nests - it is a bit unusual for you to find no sign of the nest when a predator has raided it, it almost sounds like a human. However if you look around a bit you might still find some remains.
Cats are always suspect in bird nest raids, they are among the biggest threats to many birds.
Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.