As an ex mechanic I always Reccommend get the car put through the test first to see what they fail it on, then get that fixed. I’ve often found that garages often want to fix stuff that won’t cause it to fail the test, or anything that increases the labor hours! I saw one occasion where a reputable dealership was suggesting over €2k in work on a 12 year old car (not worth it) I stepped in as it was an elderly neighbor. Put it in for the test, and €1.50 later on a stop light bulb, she had a new cert on her car!!!
As for tyres, I have used part worns to get cars through NCTs before so they serve a purpose but if you plan to keep the car, new tyres are often the way forward. There’s a New brand going around now, Black Lion tyres, quite cheap but bloody good quality. They’re advertised outside the relevant tyre shops. I put them on my car 2 years ago, they have been terrific, low road noise, good economy and most of all, good grip.