All a 'hard reset' (aka 'factory reset' - they are the same thing) does is delete user-installed apps and data. Nothing else. It won't ever undo any changes to the operating system (updates, rooting, changes made after rooting, etc - which means that it also will not rescue people who root phones and then remove things they shouldn't have). And it certainly wouldn't remove branding - if you took the name 'factory reset' literally you'd expect it to restore any branding and bloatware you had removed rather than the alternative, since that's all part of the original software for a carrier device (though, as noted, the name is misleading and it doesn't actually restore anything).
There isn't a raw 'Nexus' Note 5 to get back to though. The Note 5 software was written by Samsung, using Android as a starting point, so the version of the phone sold as unlocked+SIM-free is the 'rawest' that there is (Samsung user interface, Samsung apps, plus whatever bloatware Samsung decide to add themselves, just without the stuff that the carriers add).
As for making it a UK/EE Note 5, did that even exist? If they ever released the Note 5 in the UK it was very late, but as I remember that was the handset that Samsung's marketing decided that we didn't need (they didn't think the S-Pen was used much here so we'd be fine with one of the Galaxy S phablets instead). That was a year of bad decision making by Samsung, since it was also the year that they decided that nobody needed SD cards, both decisions they had reversed by the following year.