Hi there.
I have a few questions:
What time does he usually wake in the morning, what time does he go to bed & what time are you trying to put him down for a nap?
What time does he nap at the childminders?
Does he sleep in his cot at night-time or do you co-sleep?
Can he settle to sleep independently at bedtime or do you need to rock/cuddle/feed him to sleep before laying him down?
Have you ever established a good wind-down routine before trying to put him down for a nap/BT or has he always been a pushchair napper?
Some LO's do drop their daytime naps by 19 months but I would say they are in the minority & the majority will nap til between 2-3 years old. Since he is napping well at the childminders then this would suggest to me he probably does need a daytime nap still.
I think if he has never been used to napping in the cot it may take quite some time to do it, but it could be done. Personally I wouldn't advocate just leaving to scream in there b/c he is bound to be upset if its something he's not used to. But if you can establish a wind-down, similar to what you would do at bedtime, perhaps a snuggle, read a story then upstairs, into sleepsack, close curtains, say 'its naptime now & I'll see you in a bit' & then pop him in bed, & then use something like walk in/walk out or the checking method then he should eventually fall asleep. OK he will probably scream, but you are there to comfort him if he needs it. The best advice I can give to you though is that if you decide to try for cot naps, be consistent. It may well take more than 'a few days' before he is happy to sleep in there. Possibly a few weeks & he may also have a regression during this period. If you hang in there, he will do it. And even if he doesn't fall asleep, he will get perhaps an hour of 'quiet time' which will recharge his batteries a bit ready for him to get on with the rest of his day.
HTH.x