Tele Nut in the Right Place?

hey folks,

first time posting on here, I was hoping for some advice from you guys about an issue I have with a Telecaster.
I bought a 'Big Tex' Guitars BTG '63
(this one: https://reverb.com/item/78236-big-tex-telecaster-63-2014-lake-placid-blue-brand-new )
I live in the UK, but bought in the US, hence why I haven't taken it back to the shop it came from. I LOVE this guitar, and while it is really great, it hasn't come without it's issues. I had some truss road/fret buzz issues, but this one specifically concerns the nut and how it is placed. All the strings seem to be more or less in the right place, and the guitar is comfortable to play, but I realised, on the high E string, that I occasionally fret the string OFF the fretboard (similar to fretting out), or more obviously, I struggle to do pull offs, as the string physically goes off the fretboard. This particularly happens with pull offs to the open string. I then realised the nut isn't quite in the centre of the neck, as one side of the nut is slightly sunken into the neck while the other side (high E side) is the opposite with a minor bit of nut sticking out, which seems to be fairly obviously moving the strings all over slightly and causing this issue. I've never come across it before, but this is the first 'hand built' (so to speak) guitar I've owned. I don't know about everyone else, but I don't typically play lots of pull offs etc when testing a guitar, and it slipped through my fingers.
So, I have a couple questions,

A) has anyone experienced this before, and if so, what was the fix?

B) If the NUT was to be altered, can it just be moved, or will have to be replaced completely?

C) Is it either a cosmetic or performance altering process? (aside from fixing the issue)
i.e. should the guitar play very differently after the work?

I have been so busy recently that I haven't been able to get the guitar to a luthier, and thought I'd gather any opinions/info I can.

Thanks!