You should ideally begin the admissions process for private school as early as September of the preceding year. The best private schools are often highly competitive, so the earlier you express your interest, the better.
Check into the guidelines for your school of choice. They might run anywhere from December to February for the deadlines, but even as late as May. Most acceptance decisions are made in the spring, so even if you believe you are running behind schedule for enrollment, pay the school a visit anyway, as you may be pleasantly surprised.
Don’t forget that there could be some last minute or mid-school-year spaces become available if families are transferred out of town. So be sure to ask about the possibility even if the school is about to start, or if switching your child half way through the year is worth it for you.