As I've argued a few times, the date of Christmas being on the 25th is by no means linked to Saturnalia, or at least with very little certainty, this is a very late development in terms of speculation.

(here's a thread with /u/Gadarn; one by me; another by me).

Anyway, none of these touch directly upon your question, I'm just pointing out that this "replace Saturnalia" theory is not itself well founded.

The reason why non-Western (i.e. not Roman Catholic and not Protestant) churches celebrate it on 6th or 7th January, has to do the Julian and Gregorian calendars. The Julian calendar works on a solar year being 365.250 days, but the solar year is in fact actually closer to 365.242 days. in the 8th century, Bede calculated the error at more than 3 days, by the middle of the 13th century, this was beginning to make a noticeable mismatch between calendar and seasons, and Roger Bacon, an Englishman, noticed this and petitioned to have it 'corrected' (about 7-8 days at his time)

This didn't take place until the rule of Pope Gregory XIII in the 16th century, who entrusted Christopher Clavius, a Jesuit astronomer, with the problem. He built on the work of Aloysius Lilius. The solution was to remove some of the leap years, so that while a leap year occurs every 4 years, on the century it only occurs when divisible by 400. I.e., 2000 had a leap year, but 2100, 2200, 2300 will not.

Besides this, the Gregorian calendar 'adjusted' for the slippage between solar year and calendar, by skipping directly from the 4th of October to the 15th.

This change eventually (a) dominated European countries, and (b) their respective denominations. But although other countries did adopt the calendar change, other denominations (Coptics included), did not, and continue to celebrate events such as Christmas according to the Julian calendar. In fact, the mismatch will continue to increase by about a day every 100 years.