I live in Irvine. We moved from West LA when my son was two, largely to be near family, but also for the better schools. If you can come down here, I totally recommend it!!! I love living here - great friends, great schools, parks everywhere, dozens of supermarkets and Targets, etc.
Anyway, in most OC schools, kindergarten registration usually opens in March. However, kids who move to the neighborhood anytime before the start of the school year can generally enroll without a problem. Occasionally, a neighborhood school will fill up and a kid will have to go to another nearby school, but that is rare. And, since all the schools in Irvine are excellent, it's not really a big deal anyway.
If you're staying in LA, a lot of the best schools are local charter schools (public and free), but they do have lottery systems. If you don't get your paperwork in on time for the lottery picking, you'd have virtually no chance of getting in later in the summer, since there will still be a long waiting list. If you're coming to OC or the valley, it's not as crazy.
If you can shoot to be in your new neighborhood by spring, the enrollment process will probably be easier. But as long as you're there before the start of the school year, you should be fine.