If it is accurate to real world Japan, where Kumon-Mi is based off of, the Japanese school year starts in April. Here's the thing, though, we don't know how far into the school year they were at the start. It never mentions the first day in game being the first day of the school year. Even if it did start in April as in real Japan, it could be a couple weeks in or even a different month when the game starts. The odd seasons also don't help much to figure it out.
We can guesstimate based on a few factors.
1) Chapters end around Christmas/New Years, and a change of seasons follows. This is also likely the end of the loop.
2) Chapters span a period of two resets.
3) Per Maya, resets happen roughly quarterly, so 3 months, give or take a couple of weeks.
4) The class is familiar with Sensei at the start of the game. Enough so that his shift in teaching style is identified immediatley as out of character. We also have Makoto, who's become impressed with his intelligence. This is clearly not the first month of the school year.
I think the game starts in July, stretches over the Christmas, sometimes going as far as into the first week of January, and then skips back to July.
This gives us ideal beach season, Halloween, and Christmas. I believe this also covers every character whose birthday is actually mentioned as an in-game thing that happens (rather than a nebulous day that happens offscreen).
But the span isn't consistent. Maya acts like her birthday doesn't happen often, and chapter 1 seemed to skip it entirely.
