Sounds like he is more than ready for a short nap or no nap. I would ease into it by letting him nap for 30 mins or so most days. Personally I would not go cold turkey, I have eben finding by day 3 of no nap LO is an absolute mess!!
If he is fighting the nap --- can you work it so that he will rest in bed without fighting it? Make it out to him that it's rest time, set some firm rules about what he is allowed to do at rest time. Talk about the new rest time rules for a few days so he gets the hang of it. For example he just needs to lie down in his bed with his books, he only has to rest, not sleep. He can have an extra pillow so he can be propped up and read, but must stay in bed. That way, no matter what he is still getting down time and even a short nap. At least if he doesn't sleep he is getting some time out.
On the no nap day you should be able to bring the bedtime foward .. hopefully by an hour.
I would do the same at daycar. Tell them he still needs the down time and should rest, and to wake him after 30 mins or so.
On a no nap day, things are pretty shakey in our house by 5pm. On day 2 or 3 or no nap is is meltdown after meltdown. I bring dinner forward an hour, bath time forward, and bedtime forward one hour. We just do low key activities, like puzzles, books, pretend games like shops, playing naimals in the jungle etc. TV is used if desperate, he can help mum pepare food in the kitchen, no visitors, no outings, no running games, no rumbling with Dad, no riding the bike outside. Bascially any activites that will trigger a tantrum or frustration is avoided.