Is priority reevaluated in between each robot's movement or do we keep how it was at the start of the register?
The rules for Robo Rally can be found here.
On page 4 it says the following (emphasis mine):

Determining Priority
In ROBO RALLY, when a player has priority, it is that player’s turn. Rather than taking turns based on where they are sitting, players take turns based on where their robot is sitting on the board.
At any given time, the player whose robot is closest to the priority antenna has priority. After that player takes their turn, the second closest player to the antenna has priority and takes their turn, and so on.

This means that priority is reevaluated after each player's turn.