Was there any year in his childhood when there was a heavy fall of snow like that?

Good question! I checked, and the Decembers between 1920 and 1926 were all uneventful - but I can confirm that in 1927 (when he was an awkwardly disyllabic 13) snow really did fall for a good week, beginning on Christmas Eve.

The Dylan Thomas Society (I'm sure there must be one) shall hear of this.