There were eight major festivals in the Pagan year, one of which was on 20th / 21st December, the winter solstice. Surely, they could have decided Christmas Day would be on one of the other seven festival days so as it didn't clash with New Year's Day ?
Foula, which is a remote island off Shetland with under 40 inhabitants, preserve many Norse traditions of folklore and festivities. They still follow the Julian calendar for their Christmas celebrations, with all the islanders gathering in one house on 6th January. New Year's Day falls on 13th January which means there are still only 6 days between them !
The truth of the matter is that December 25th, Christmas Day, was probably adopted because it is exactly nine months from 25th March. Scholars think it possible that, because of 3rd Century beliefs, the crucifixion of Jesus took place on the anniversary of his conception. As this date can be worked out from Bible texts, because of the way Passover is kept, it was thought Jesus was conceived on March 25th and the Church still keeps that date as the feast of the Annunciation. So for Christmas they just added nine months !