We say 'at Christmas' in the UK when speaking generally, not 'on' e.g at Christmas people get together and celebrate with their families. We would use 'on' if talking about a specific day e.g On Christmas Day, I visit my cousins.

at means in a location or place "at the store" "at the park" while on could mean at a certain time or date, and also on top of something(on the bus, on the bed etc.). On Christmas is the correct way to say it. You could use on in other ways like "On Friday I..." "I have the day off on Tuesday" At can also reference to a time like "At 3 o'clock I have an appointment"