I think it depends on the child.
My dd started secondary in September. The first week she continued going to my parents on "my days" but she decided she wanted to try coming home and starting her homework instead. She gets home about 330 and I get in anytime from 515-545. She often takes herself to the junior gym session at the local sports centre after school as there aren't really any clubs etc at school she is interested in.
Dd is quite happy being left, but she is wise way beyond her years. I also have a good support network locally.
My next door neighbour is housebound and very much like a sister to me so dd knows to go there if worried. My parents are at the opposite end of the village and she is able to walk there or call them if something worries her. My mate whonisnlike a brother is also just down the road, he finishes work at 3 and is home by 345 so is olalso on call if needed.
When you say you will be moving, will it be within a similar area or somewhere completely new?
How does the suspected ASD display?
There are lots of things to consider, I don't think it is as straightforward as "he is 11 it will be fine".
It is very much about individual circumstances and the individual child.
I have dsd and dss via dd's dad. Although we are not together they will always be my dsc. However, dss was only just capable of being left alone at 16. He was nowhere near ok at 11. Dsd probably OK from about 13.