You need to ignore the score. It’s just what that individual credit agency shows for illustration, if doesn’t mean anything else. All the companies you apply to will do their own scoring.
He needs his reports from all of the main three - Transunion, Experian and Equifax.
His debt won’t be linked to his mum if he hasn’t shared any financial accounts with her. Address blacklisting isn’t real, and hasn’t been since well before I started working in finance. If he is linked to her through credit accounts, she’ll show as an associated person on his reports.
Right now, mortgage lenders are being quite cautious; and if he’s been on an agreement to pay a credit card, that might be enough to make them cautious. That means he couldn’t afford to make the payments. Paying the debt off completely might help with affordability but won’t have a effect, as it’s only £1k, and it’s probably not going to affect the fact that he was on a payment plan... who was that with? Typically the big usual credit providers won’t entertain a payment plan unless there has been missed payments/defaults. If it was sub-prime credit already, that’ll make a difference too.
You need to buy his reports from all three places, and look at his actual reports, not his score. It takes six years for things to drop off your report, so all the debt he has had in the last six years will show, but typically anything that’s old (4-6 years) will be less of a concern than anything recent (1-2 years).
You also want to consider that buying a house together will make you financially linked, and his score might affect yours. My ex partner had an okay report, mine was pretty good. As soon as we had a join product, mine tumbled and it took two years to get the credit reference agents to disassociate us after our financial link ended. This might be relevant if you’re planning to buy anything else on credit.
A broker will be able to tell you what mortgage providers will accept you with your files as they are, and they’ll have an even better idea if you send all the files. It’s possible that it’ll need to be a subprime lender if the payment plan is recent, so you may need to budget for a higher interest rate.
Best of luck!