'Edit' tab does not appear on user's account page? On my site when a user navigates to their account page there is no 'Edit' button for them to be able to edit their details. From the documentation online it seems this should appear automatically.

Is there any way this could have been disabled by accident or are there any modules that could have interfered with it?

A user is able to edit their profile if they add /edit to the link to their profile page, so it is not that they don't have the 'edit' permission - they definitely do and are able to edit their details successfully if they go to this link.
Oceanescence, some times people have disabled or moved the tabs block. If you are using a custom theme try switching to Bartik and see if it pops up. 

If not try digging into the permissions. Just log in as Admin 1 and see if the tabs show up.