My follower won't pick up items in the Nintendo Switch edition of Skyrim.
Do you have any customized controllers, particularly the activate command? If so, this reddit thread mentions this might be a bug on the Switch version of Skyrim. Followers do not seem to pick up items unless the activate command is the default configuration, that being the A button.
Try resetting your activate command to be the default. A user on the reddit thread confirmed this was a solution for them.