My general advice is to remove it. If you get the beep for secondary or at the discretion of an officer for the body scanner or a pat down, then it just creates needless hassle.
As far as actually alerting at the WTMD, It depends on the tolerance of the metal detector. The overwhelming majority in the US won't trigger for a watch, even if it is metal. Neckklace would depend on the size and metal content.
One metal detector I am aware of with an insanely low tolerance is the WTMD at IAH Terminal A Precheck lane. That would trigger for both my shoes (leather with no visible metal) and my belt (only metal is a small buckle, rest is leather). I would go through there once a week for about a year and people would see me taking off my belt, shoes, and taking out a laptop (I had two) and lecture me on how to use the lane properly
(In contrast, I had to still remove one laptop at IAH Terminal C, but the same belt and shoes did not trigger the metal detector there, or dozens of other US airport checkpoints).
Like Often1 I just put these key items into a pocket of my carry-on bag and put that in the X-ray before walking through the metal detector.