Firstly, I don't think any Christian calendar (there are many!) starts with Christmas day - most start with the Annunciation or the beginning of Advent.

Remember that there are several different dates for Christmas Day depending on what denomination someone belongs to, and several different dates for New Year. Many Middle Eastern Christians will start the liturgical year at the start of December and celebrate New Year well into spring for instance - spring is a more common time for New Year worldwide, January 1 as New Year's Day is a Roman concept and many Christians worldwide use entirely separate calendar systems. However, essentially, New Year goes by the civil calendar and Christmas by the Christian religious calendar - and the latter has been absorbed as a secular celebration in Christian-majority countries. Remember that many Christians worldwide live in countries where Christmas Day is not a public holiday and where Christianity is not the main or even a major religion.

Sources: Thomas J. Talley, The Origins of the Liturgical Year (Pueblo Publishing Company 1991 ISBN 978-0-8146-6075-1)

The Orthodox Study Bible. Nashville, Tennessee: Thomas Nelson Publishers, 1993 (Orthodox Christian, not Orthodox Jewish)

Pathikulangara, Varghese. Mar Toma Margam, (A New Catechism for the St. Thomas Christians of India), Kottayam: Denha Services, 2004

Anthony Aveni, "Happy New Year! But Why Now?" in The Book of the Year: A Brief History of Our Seasonal Holidays (Oxford: Oxford University Press, 2003)

Hope these sources are adequate.