DD3 came home upset yesterday as she was told off for not wearing a coat for playtime.
I believe that, at a lot younger than 8, a DC can make their own decisions about comfort levels. It is absolutely nothing to do with the TAs what the children are wearing, as long as they are complying with the uniform rules.
DD2 also mentioned that the same TA challenged her about not wearing a coat, and offered to check whether DD2 was telling the truth when she said she hadn't brought one to school. 
She won't allow them to remove their coats if they're boiling because of playing football either.
Surely, rather than nagging them to wear a coat, it would make less hassle for the TA to let them be independent and choose their own attire? The DC could always nip to the cloakroom if they have made the wrong decision.