We have a child in near identical circumstances. He has been going outside in the snow in just socks, jogging bottoms and a long sleeved t-shirt! We have let him go outside as he is making the choice what to wear and he can come in whenever he wants. We did have wellies and a coat on standby if he chose to put them on and asked mum to provide a change of clothes. It is very difficult though I agree!!!
In regards of coming into school we have found spending the first portion of the morning doing a chosen activity (a game or craft activity) has settled and made them look forward to coming into school.