I have adopted from Cats Protection. I found a kitten on their website, which I made enquiries about, and visited at the fosterers home.
The home visit was just a look at the area where we live, (any busy roads) the home (enough room, who lives there including DC, any previous pets and existing pets, experience with cats) and garden, (access from the home, cat friendly, shade for hot weather and shelter from rain and snow, places to hide and things to climb) They aren't looking at how clean and tidy the home is.
That all sounds a bit full on, but they are just establishing that you don't live in a bedsit with 12 existing cats and no garden, next to the M1! The fact that you've had cats before, and that DS has been gentle and loves them should be a bonus, CP don't usually re-home kittens in families with under 5's, but if you want a cat it should be fine. DS will be able to visit the cats you are interested in, and I think it's pretty well known now that people with autism have bonds with animals that they don't even have with people, often beneficial to both. You may find that a cat chooses him! You may need to visit prospective cats at their fosterers homes, as well as at a rescue centre. You can visit those cats that interest you by appointment.
When you have decided on a cat, you have to do some form filling, and pay a donation to cover CP's costs food/vaccinations/microchipping/spaying, I think it was £50 for my kitten 5 yrs ago, and you can pay more voluntarily. I found the fosterer of my kitten was really generous, she gave me food and toys, and the vetbed the kitten slept on, lots of care leaflets and the vaccination record card, and the microchip details in a folder.
i hope that helps, and that you find your ideal cat soon!