He's only 2! This is when you can start edging into introducing him to the potty, not expecting him to be full trained. I can see a preschool requiring potty trng, but a daycare? Keep shopping, are they kidding? If they don't want to change diapers, don' run a daycare.
Unfortunately, two is really early, on average, for a kid to be even pee potty trained. The average is about 3. I was 18 months (according to my doting mom) but my brothers were 3 and 4. I was very unusual, and my brothers were, well, boys!
And the more you push, the less luck you'll have. It will not happen until he is physically ready and he has to WANT IT. If you have a stubborn kid, good luck! When (key word!) he is ready, tell him they don't let big kids in the pool, for instance, that still wear diapers. whatever he's looking forward to, tell him he can't do with diapers.
My daughter was wearing pull ups for awhile but then she started wetting them regularly like she would a diaper. (Thus my agreement with Supernanny that pull-ups just confuse the issue.) After a couple of days of that nonsense, I said fine, lets do the padded "big girl" undies and see if she likes that...which she didn't! She was still over 4 before she was fullly trained, #1 and #2.
Every kid goes at their own pace and there is not much you can do to speed up the process. Give him more time.
-S. K
P.S. This whole pushing kids into what they're not ready for yet is really making me ill. Like, kindergarten is what 1st grade used to be, preschool is pretty much a necessity...my mother in law was a preschool teacher for twenty years and she said education has gotten really age inappropriate. The potty trng expectation is in the same category. And then we wonder why they get into things they shouldn't be doing yet when they're teens. Okay, rant over.