Here’s a scatter graph of temperature and precipitation anomalies since 1910 for the whole of the UK using the 1910 gridded climate data series from the Met Office. The opaque green square at the centre surrounds any winters that were close to the average for temperature and precipitation.
The one thing you can say about this correlation analysis is that if the coming winter is going to be cold, then it’s also more likely to be drier (38.5%) rather than wetter than average (8.3%). Conversely if it’s going to be mild winter there’s a greater likelihood statistically that it will be wetter than average (24.8%) rather than drier (14.7%). Interestingly there’s never been an extremely cold wet winter in the last 108 years in the UK.