Funnily enough the new year did use to start in spring before the adoption of the Gregorian calendar, so you were almost right, just 300 years too late
Interestingly, I did learn about that later as a kid, which stuck with me for a completely different reason: the new year used to start on my birthday! (March 25th)
New year was moved to january in antiquity, nothing to do with Gregory XIII or his calendar.
Not in England, Wales, Ireland and Britain's American colonies, which all made the change of the new year from March to January in 1752 alongside the switch to the Gregorian calendar.