That Gov.UK list is for only very minor faults, and my interpretation of the link doesn't include tyres or an alledged 'corroded' rear spring.
James' argument is that the tyres are legal; theirs is that they are illegal. Whether or not they are actually legal or otherwise, their stance is such that as they were allegedly a 'dangerous' fail, they couldn't permit the car to be driven on a highway, and therefore get the job of replacing the tyres.
If James is happy that the tyres are legal, he should be able to take it away, repair the defects, and return it within 10 days for a partial retest, and probably have to pay the retest fee. Many smaller garages don't charge for this, but the way this has gone, they are not likely to waiver the partial retest fee.
This has probably been said previously, but as this topic is now so long, I'm not going back through it to check.
I think there are several morals to this - preferably use your local previously known MOT man, don't use franchised dealers who are likely to try to get 'up sell', and don't fall for the 'free MOT for life' or 'MOT's only £25' marketing rubbish....