Welcome to the club and thank you for doing a little research ahead of time.

I suspect that most of us will have the disadvantage of not having a Casino Royale schematic available (unless you can provide one here or on ipdb.org) but here is a similar circuit from Williams Little Chief from a few months before:
Little Chief Coin unit (resized).jpg

The way this circuit works is that when you drop a coin or press the Replay button, the Coin relay activates which closes the leftmost switch above. The Coin relay also starts the Score Motor which turns through one complete cycle. At some point in that cycle the 1B switch on the Score Motor (2nd from the left) closes briefly which sends a pulse from left to right through the schematic above. That pulse will go to either the Coin Unit Step Up solenoid or the Coin Unit Reset solenoid based on the position of the Make/Break switch on the Reset relay in the red box. When it's working properly the Make/Break switch should only let the pulse get to the Coin Unit Reset Solenoid when the Reset relay is active (when starting a new game), otherwise the pulse should default to going to the Coin Unit Step Up solenoid (when adding another player).

You could verify that the pulse is getting generated properly by watching the Total Play Meter which should advance each time you press the Replay button or drop a coin. If that's working properly it's likely that either the Make/Break switch isn't adjusted properly, or that the Reset relay is firing each time you add a player.

Your schematic may differ from this one but perhaps it's similar enough that you can draw some clues from it.

/Mark

I'd stick with the Reset relay Make/Break switch for the moment. The fact that you're getting a pulse verifies that two of the three switches in the circuit are working correctly.