There’s no hard and fast rule, not a date anyway. I know they can ‘track’ Spring though.
Personally I follow the meteorological seasons, Winter = Dec, Jan, Feb, Spring = Mar, April, May, Summer = June, July and Aug and of course Autumn is Sept, Oct and Nov.
The other method calls the 21st of Dec as the first day of winter but it seems a bit odd to be to call the darkest day of the year as the start of winter in the same way the lightest day of the year as the start of summer.
The seasons, especially in the UK though are effected by a lot more than just the position of the Earth relative to the Sun and they don’t always play ball with the calendar, as for now – we’ll a sudden drop of 5c from last week and a bitter wind has made it feel very winter like around here, but it’s still 10c at midday, it’ll get a lot worse before it gets better. Late Autumn, Early Winter – does it really matter?