The easiest way to sum this up is to say – if your electrical appliance turns off when you unplug it (take the power away) then it’s mains powered and thus needs a PAT test.
If it stays on then it is battery operated, and doesn’t need a PAT test (but it’s leads or chargers do)
If it doesn’t have a plug, and now is wired into a ‘spur’ that has a switch to switch it off (effectively unplug it) then that also needs checking. If the spur doesn’t have a switch, it may also need checking. However, with these, we only check to the plastic plate of the spur, never inside it, as that is an electricians territory. See Fixed Appliance Testing for more info on this.
If it was never intended to have a plug, like a shower or an escalator, then it’s nothing to do with us.