I had this happen twice, both due to my manually triggering a birthday for a toddler. (The first time I was playing on a long lifespan and manually aging the kids through their entire childhoods, and the second time I had quit right before 8:00pm, and I think the game "forgot" that it was his birthday.) Each time, the child's school "shift" began in one day and change, instead of one hour or whatever, and each time the kid went to school normally the next day.
If your sim is ready to go to school on the second day, but you don't want him to miss the first day, you can use nraas to make him go. Click on the child, nraas>intermediate>career>go to school and the action will appear in the child's queue. The mod assigns a shift of six hours for children and five for teens, just like for a job, but the "shift" starts to be counted from the top of the hour when it was assigned. So if you give the command at 8:59, a child will be in school until 2:00, not 2:59.