Can I Time my speedrun with a stopwatch on my wrist watch since it is an external timer?

you can but to Submit you will need to Record or stream a video with a timer as per the Rules.

"Timing starts the moment 'New Game' is pressed and ends the moment Wander falls into the pool after defeating all 16 colossi.

Runs use the Real Time Attack (RTA) timing convention, meaning runs are timed using an external timer (such as WSplit or Llanfair) which can't be paused during the run."