I linked one example—a video on Christmas trees from a religious studies scholar. They have similar content on the date of Christmas, and there are plenty of sources on each other tradition.
Here's another one on Saturnalia from the same scholar:
One small exception
Russian empire turned a figure from Pagan mythology into a St Claus like figure
https://en.wikipedia.org/wiki/Ded_Moroz
But my understanding is that him handing out gifts didn't come from Pagan traditions and was a result of largely copying Santa Claus traits.
Later during Soviet times secularish Christmas traditions including Ded Moroz were moved to New Year's.
Here's another one on Saturnalia from the same scholar:
https://www.youtube.com/watch?v=5lsctaPJSvo