What should I do

I have to go back to work. My son is 3years old and 4months.

He has been slowly settling into a nursery school.

He usually did 8:30 to 11:30 and would come out exhausted and fall asleep in car.

His times are being increased 8:30 to 3:30. The problem is he comes out way past being exhausted and possibly to late for a nap. I will put him to bed at 6pm but due to him being so over tired he is waking constantly throughout the night and struggling to fall back to sleep. Last night was absolutely hell. I put him to bed 6:30 he woke at 8 I managed to settle him he then woke up at 10:30 and even with me laying next to him he couldn’t fall back to sleep. Finally at 2am he fell asleep woke once again at 5am but fell back to sleep to 7:30.

He is now a very grumpy and teary little boy. How will this work when next week he will be doing 5days 8:30to3:30

Should I let him nap at 3:30? If so for how long and what bed time!

A lot of people say he shouldn’t need a nap at his age but he literally doesn’t seem to function without one!

No options to nap at nursery and very unlikely that he would ever nap somewhere other than car or home.

What would you do? What should I do