Okay. Here is my answer. According to Bible, New Year does NOT occur in the winter month of January. It rather occurs in Spring in the month of Nisan (March-April). This was precisely when the Early Christians celebrated new year. So how did the New Year "change"? Read on

Until 1752, Christians of England and US celebrated the New Year on March 25. "The year beginnith on 25th day of March"

When Christians celebrated New Year on March 25 Who celebrated the New Year on January 1? That is right. The pagans celebrated the New Year on January 1. They called it the festival of Calends. They decorated their houses with wreaths and offered sacrifices to Goddess Juno

Initially, the Christians banned the January 1 New year festival. "The whole wicked celebration is Pagan" they said.

January 1 was banned by Bishop Eufronius in the council of tours (567 CE). The Christians continued celebrating New Year on March 25 "God created the universe on March 25"-Pope Benedict

It was only with introduction of Gregorian calendar in 1582 that January 1 was reinstated as New Year. By 16th century, there were no Pagans in Europe celebrating calends. Having a new year on March 25 in a Calendar starting on January 1 had turned out to be very inconvenient

Yes, the Gregorian calendar is Christian. Yes, the shift involved a change to accommodate Easter. But this has got nothing to do with January 1 which was the new year directly adopted from pagan Julian calendar

The feast of circumcision originated in 7th century. It was not recognized by Roman Catholic Church until 13th century. What has this got to do with the New Year which was celebrated by Pagans since 150 BC and which was banned by church?

Also, the churches that celebrate the feast of circumcision are mostly orthodox churches. They still follow Julian calendar and have not adopted Gregorian calendar. So, the feast of circumcision falls on January 13 according to our (Gregorian) calendar & March 25 is the new year

Which means there is no real correlation unless we intermingle both the calendars. When the "shift" happened, it was not only January 1 that was shifted. All the days were shifted. Hence, all the days remained relatively the same