What can your son do, swimming wise? If you can give a bit of an indication I might be able to say what might help as a next step.
This is a bit of an essay, sorry, but it's to give you an idea of what order things might go in.
My DS 9 (AuDHD, would probably get a DCD diagnosis if he didn't already have an autism dx) took forever to learn, but in the past 6 months, aged 9, has gone from the 4-5 year old level of swimming lessons, to a level with the other not-very-competent 8-9-10 year olds.
He has been having lessons since age 5, a mix of 1:1 and group. The thing that made the most difference was 2 lessons a week, not whether it was 1:1 or not, and him actually developing core strength. He is very tall for his age, very thin, very weak generally.
At age 4 DS could kick with a kick board, doing (very very closely supervised) lengths of a 15m pool. He could put his face in and blow bubbles as well, but generally chose not to. Until about age 8 he absolutely would not swim by himself, with or without a kick board, so it was very difficult to get him to practise anything at all.
From age 5 to age 8 he could swim about half of a 15m pool with arms stretched out in front, face in, legs kicking, but couldn't roll to the side to breathe, so had to stop and put his feet down whenever he needed to breathe. He also couldn't float at all, so couldn't begin backstroke. I persevered with lessons even though he made no progress for years.
At age 8 we switched to two 1:1 lessons a week, and he began to get the hang of side kicking, but couldn't link it to freestyle strokes. He also began to get back kicking (preparation for backstroke).
At age just 9 we took him on a 2 month backpacking holiday where he carried his own pack. Not hiking miles a day with all his gear, but enough walking with a proper hiking rucksack (i.e. weight supported mostly on padded waist belt), with a bit of weight in it, each day, that he developed abs. Instant difference to the swimming. You might not need a backpacking holiday, just a change of school bag!
When we got back I put him in 2 group lessons a week, and he suddenly could link 3 freestyle strokes to sidestroke to breathe, then 3 more freestyle strokes. It took him about a term to not get muddled over which side he needed to roll to. He could also do backstroke with no difficulty. So he can now do 25m freestyle and 25m backstroke.
A year ago I'd have thought this was never going to happen. Now he's the worst in the group at breaststroke, but I have slightly more hope now!