Thank you all so much for all the help and the really useful replies!
i have some good news, been really busy over past few days trying to sort this out in addition to working full time. i took everyone’s advice and I sent the seller a letter threatening small court action (I found a template on the citizens advice beureu website) and like magic, he called me on Wednesday and agreed to get it fixed!
with regards to the humming noise, I took the car to a 4 wheel tracking place on Monday to get it checked out and they did an alignment on it but couldn’t identify the reason for the humming, they did however tell me that my aftermarket roof racks are making a lot of noise and I since realised that it’s a lot worse when the sunroof is open! That’s really reassuring to me but the last thing I’m going to do is get all the wheels balanced, they have Bridgestone SE031 Potenza on all of them I think it’s the OEM tyre.
with regards to the main issue of the engine light, I showed the seller all the fantastic replies and advice about diagnosing the error as he was adamant it’s an easy fix of switching the sensor. I’m a bit concerned because I took it to the garage he recommended and they were unsure where the sensor is exactly on the car and which one it is. I was in the mindset of beggars can’t be choosers so I left it there for the mechanic to diagnose and repair at the sellers expense. Place isn’t where I would personally take my car, called S K motors on Windham Road in Bournemouth.
I’m hopefully collecting it tomorrow. Just hope that it’s fixed and if a sensor is replaced they don’t use a cheap one.