So now is there anything I should do? For example, should I keep calling or emailing the council to ask about the progress? Should I get in touch with the school itself, hoping they will remember my son's name?
No, please don't. It will make no difference whatsoever. The Admissions team will contact you again in the event that a place comes up. The school has no part to play in the process and they can't somehow "forget" about him.
All you can do is what the name says - wait. If another child leaves, and no one else has moved into the area who is a higher priority under the Admission rules, your son will get a place.
If you hear nothing further in a couple of months, you could call the Admissions team, just to check that he is still in first place on the list, but normally they would write to advise you of any change to that.
(You could appeal for a place, but I wouldn't advise it. Infant Class Size appeals are incredibly hard to win, and you would need a lot more on your side than just "we really want him to go there".)