I have had a look at the most frequently used climate indexes and how they correlate to the mean winter CET over the winter periods 1950-51 to 2019-20
What comes out strongly is that the only indexes which have a strong correlation to the CET are the Arctic Oscillation (AO) and the closely related North Atlantic Oscillation (NAO). When the average of the winter values of these indexes are used, the correlation coefficient to the mean winter CET is 0.67. No other index gets higher than 0.19 which may be sufficient to provide “nudges” to the mean winter CET but are far from influential when taken just on their own.
Since the AO and NAO are closely related (correlation=0.78) only the AO is used for the rest of the analysis.
In its positive index phase (high polar vortex strength, low height anomalies over the north pole) the AO leads to a strong “zonal” jet stream across the Atlantic and generally wet and mild conditions in the UK. The opposite occurs with a negative index leading to a weaker flow and more “meridional” or buckled jet stream and often a cold “blocked” winter
At the same latitude in different parts of the world the effect of the AO on winter weather is different. In the north east of North America (such as Newfoundland ) there is a reversed correlation with the coldest weather matching the positive index. In the north west of North America (such as Vancouver) there is no significant correlation. In Eastern Russia the correlation of positive AO index to milder winters starts to return and in Eastern Europe (Minsk for instance) this correlation is as marked as it is for the UK
It is known that the Jet Stream anchors itself to the Rocky Mountains in Canada and, for a strong flow in a positive AO Index, dives south while crossing the USA so that the North East of the USA is on the “cold” side of a powerful jet with winds blowing from the interior of the North American continental mass. Hence this part of the world has the reverse relationship with the index to everywhere else. A more variable set-up would be in place here for a negative AO index with a meandering jet and some easterly flow from a relatively warm ocean.
For the Eurasian land mass a strong westerly flow will bring milder weather from the Atlantic across most of the land mass
It is worth noting that in northern Eurasia the AO index has the most significant correlation with winter temperature but in the north east of North America it is the AMO index (0.57) and in the north west of North America both ENSO (0.36) and PDO (0.47) are much more significant
The frequency of winters with a positive AO index (mild in Europe) has increased noticeably from an average of about 25% in the period 1950-1989 to about 60% in the period 1990-2020. As well as general global temperature rise of about 0.87C since 1950, there has also been a more than doubling in the frequency of positive AO index “mild” set ups leading to an overall rise of 1.66C in the mean winter CET on linear trend since 1950
The bad news is that even with all these correlations, just like the CET itself, there is no way to forecast what the AO index for the following winter will be with any accuracy. The synoptic forecasts provide a short -term forecast in exactly the same way that the temperature itself is forecast but only for 14 days ahead at the most.
There is a low correlation between the Autumn AO (Sept to Nov) with the AO Index for the following winter (Dec-Feb) of only 0.27 and a negative correlation between the Autumn AO Index and the Winter CET of -0.15
It is probably a statistical fluke but for the last 6 years the sign of the Autumn AO averaged has been reversed in the following winter average.