I have a Series 4 Apple Watch, but I cannot seem to get the screenshot feature to work. The steps I have taken are:
When I try to take a screenshot, both the watch and my phone vibrate, but the photos do not appear in the photos app on my phone.
My iPhone OS is iOS 12.3.1 and my Watch OS is 5.1.3 (needs to be updated)
Oddly enough, just waiting has resolved this issue for me.
It seems to have fixed itself.
I had the exact same problem (though with Series 8 Watch paired with iPhone 13) and spent hours with Apple support trying to resolve the issue.
In the end I discovered by chance (during a demo to the remote support engineer via camera) that pressing and releasing the two buttons really quickly made it work... holding down for even half a second before releasing, and it doesn't work... with a longer press/release, sometimes the watch and phone would both vibrate (as if it's doing something) but then no screenshot would appear on the phone.
But ensuring a very quick press/release made it work (watch screen flashes, which it wasn't before).
Why can't they make this clear in the guide, and why did nobody from support know to suggest this? Certainly, on the iPhone itself, you can hold both buttons down for a while and then release, and the screenshot is effective still... but on the Watch (my Series 8 anyway)... it needs to be a very quick press/release.
In the Photos app on your iPhone:
Albums -> Screenshotsfolder? They should be there, it works for me.