Recently saved photos not appearing in Photos app on iPhone

It may take a while, until a newly saved photo will appear in all collections.

Also, Photos you are saving from outside sources may be imported with a different capture date than you expect.

If a photo seems to be missing, scroll through the All Photos view and check, if it will be shown with an earlier date.

You may also check the filter, if it is set to "All Photos". By default, you may only be seeing a curated collection or the screenshots will be omitted, or similar, see this page in the user guide on how to check the filters and how to sort the library by the "Date added": Filter and sort the photo library on iPad - Apple Support (MZ)

Cheekychinky wrote: … the image will show up in a photo picker within another app, but it won’t display in my main Photos feed.

As léonie says, the picture may just not be in the place you're looking. A good place to check is in Utilites>Recently Saved. Utilities is near the bottom of the screen,

Sometimes menus' locations and orders are changed with the Customize menu. To see all the choices in Utilities, you may have to click the Utilities > arrow to get this full menu:

If you need Recently Saved often, you can move it up the list. I've put mine Recently Saved just under Favorites.