If you go by history, the best chances for a cold Winter is an unsettled September but there have been settled Septembers followed by cold Winters. You don't have to go back far to find an example of such - 2009.
For the sake of this, I'm gonna use the England & Wales Precipitation (EWP) and Central England Temperature (CET) datasets because they have data going back to the period I want to look back to. Let's look at the EWP for the Septembers that preceded notably cold Winters or Winters that stand out to people (back to 1870) and the CETs for those Winters. Remember that the 1981-2010 EWP average for September is 77.2mm whilst the 1981-2010 CET average for Winter (December to February) is 4.5c.
In our sample space here, I have chosen 49 different years and out of 49, 27 of those years contained a wet September (for the EWP anyway, there would be some wetter and drier Septembers elsewhere like in Ireland or Scotland for example where 1985 was wet compared to the EWP series) which is just over half of the number of the years in the sample space so there's not really much correlation with an unsettled or settled September for the EWP and a cold Winter though it is interesting, three of the most severe since 1940, 1946-47, 1962-63 and 1981-82 were preceded by wet Septembers.
I looked at unsettled Septembers for last year's Winter and I found there is an increased chance of a colder Winter for 2017/18 because of this. We had a colder Winter indeed so that turned out rather well.
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