I’m no expert in this, but I’ve recently changed doctors from Sheffield to Buxton, so both within England.
The two different surgeries use different web portals for their services, but both tie back to my NHS Login.
This seems to be a desk/laptop based access and covers all of the UK. Sheffield Doc used they own system, and Buxton doc uses something called Patient Access, a 3rd party service; I can use this for both appointment booking and prescription orders as well as other services.
Clicking on that link and entering my postcode just shows linked chemists for prescriptions and nothing about doctors’ access. Check your own local GP, they may use something similar.
From what I can make out “NHS login” (Welsh link) is a national account/service covering all of GB (Not sure about NI)
If registered with a GP in England, I can download, login and use NHS App (England specific), or Patient Access, which itself then accesses my NHS Login account. NHS app (England) can use fingerprint authentication of my phone etc.
Wherever I live I can access “NHS Login” via the web, but can also access my own required services via NHS App & Patient Access.
Your link to NHS Wales App, seems to duplicate my NHS App, and requires registration with a GP in Wales, as well as an NHS Login account.
I’m assuming you already have NHS Login from your time in England, so NHS Wales App should feed into that, along with any system your local GP uses.
Bill