Welcome! I'm living in Chelmsford, Essex.
Regarding renting, when I came over with no UK credit or employment history, I found agencies wanted me to pre-pay the 6 months of rent on a 6 month rental. Ended up staying with a friend instead.
For the medical stuff, that was one of the easier things to do, I just went down to the local surgery with my passport and registered. It should be even easier for you as a dual and much easier than the US medical system with insurance cards and this doctor doesn't accept that insurance, etc. Then, make appointments and have a chat regarding your son. A friend's child has ADHD and it seems to be well-managed for in the local school system here.