New Years Day has been celebrated on various different dates in various different places- and still is. Our current local calendar- named for Pope Gregory the thingummywhatsit- wasn't adopted until the 17th century. Slippery customer, Time; all our attempts to peg it down are a bit make do and mend. Leap years for instance- they're not an elegant solution. But then neither were the intercalary months of the old Julian calendar.
January is under the tutelage of Janus- the Roman God of doorways and beginnings and endings and all that sort of thing. He has two faces- one looking back and one looking forward, which means he has all round vision- as a God should. Perhaps the association of the month with this particular god is why it got shuffled to the start of the year. But, then again, perhaps not. According to something I read, again in Wikipedia, the month's original patron was Juno- the Great Mother. The more you trowel about in the dust the more you find that almost everything about the human past is contentious. I mean, if we can't be sure who killed Kennedy how can we possibly know what was going on in the minds of a bunch of Romans three thousand years ago? Actually, perhaps they weren't in agreement. Perhaps some of them thought January was all about Juno and some thought it was all about Janus- and some just weren't sure.