Remember, it's not a "start moving ships" button, but an "end turn" button. The end of the turn *includes* moving all ships that have moves remaining.
Suppose you have Scout1, and you've told it to go over there somewhere on autopilot. The first turn when you do the autopilot command, it will use its three moves. Then when you hit "end turn" it has no more moves (and assume you have no other ships for the sake of simplicity) and it just sits there. But if you click on it *after* hitting end turn, you'll see that it's got three moves remaining. You can over-ride the autopilot and move the ship either manually (say, arrow keys) or reprogram the autopilot by sending it somewhere else. Suppose you do neither of those things; the ship carries its three moves into the "end turn" processing, where Scout1 (and any other autopilot ships with turns remaining) is moved according to your instructions.
You don't see Scout1 move after the first time you hit end turn because the first time you hit 'end', the autopilot had already moved the ship when you set it, and there were no moves remaining in turn1. The second time you hit 'end', you're telling it to use any remaining turn2 moves.
Note that one of the configuration options will enable "show autopilot" or something like that. In that case, you will see Scout1 take its Turn2 moves before control is returned to you at the beginning of Turn3.