Try Proximity sensors, such as those made by Omron (and a lot others).
Works like a miniture metal detector.
What material is your product and what size is the screw?

You get small sensors with a 2mm sensing distance, so if you adjust the
sensor position that it is more than 2mm away from the product (If the
product is metal, otherwise no matter) but with a screw underneath it is
less than 2mm, you will get a signal.

Quentin