Mo2, if you're really not sure that your son will be going to the private school, I would be inclined to leave him at his familiar nursery.
If you feel he needs something more challenging, could you have a word with the nursery staff and see if they can help him with simple work sheets, etc. At my son's nursery they do this, but are child lead. If the child shows an inclination to read and write, they will sit down with them and give them some one to one help, if the child isn't ready for this, they are free to play with the toys.
Also, your son might be ready for some extra classes, things like junior football, dance, drama, swimming or music at weekends. As well as stimulating him, this would help him meet other children his age and make new friends if his old friends have left nursery.
Another possibilty - you could suggest at a parents meeting at your nursery, (it's happening with our nursery) is that parents pay a little extra in order for a specialist teacher to come into the nursery every week and give the children a lesson.
The behaviour you describe sounds typical for my son, too. He's nearly 4. In our case I don't think it's anything really to do with the nursery, it's just normal 3/4 year old behaviour.
At the very least, could you wait and see how your son is behaving after a few weeks into the new term? if you still feel all is not good, then make the decision to move him.