Hiya
It sounds to me like there are a few issues going on here.
Firstly there is the matter of him not falling asleep independently at home. He can clearly do it since he does at the CM's. Its just a matter of what he is used to. Now is this something that you are wanting to change? Do you want him to sleep independently at nap & bedtime and in his own cot or are you wanting the co-sleeping / BFing to sleep to continue? In all honestly, while they continue, you are more likely to have NW's b/c DS will not know how to put himself back to sleep without your help. But obviously this is a very personal decision and one that I'm not going to push. So let me know your thoughts on this & we'll work out an approach to suit you.
Secondly I would really recommend that you implement a regular routine for him. Babies thrive on routine, it helps them feel secure to know what's coming next. And of course a well rested baby will be a happier baby.
He sounds like his sleep needs are changing if he is no longer wanting to nap at his old nap times, but I'm not entirely convinced the answer is to completely skip his morning nap yet as he is still quite young (most babies do this at 15-18 months) & it sounds to me like the reason you/CM are struggling to settle him in the morning is because he isn't tired enough at 8.30/9am, and the reason you are struggling to settle him in the afternoon (even when he hasn't had a morning nap) is because he's overtired. An overtired baby is always very difficult to settle because they get so wired they simply can't switch off until the point of exhaustion.
I wonder if you could first of all try getting him down for a nap in the morning at 9.30am, instead of 9am, and see what happens. Try for a few days & if it doesn't work, then perhaps try at 10am for a few days. What you need to do is see if you can catch him when he's nicely sleepy, before he gets overtired, as he will be much easier to settle. Let him nap for 30mins and then wake him, and then try and get him down for a PM nap around 2.5hrs later. So your routine might look like:
Wake: 7am
Nap: 9.30 - 10
Nap: 12.30 - 2.30 (let him sleep for 2hrs max)
BT: 7pm
If you want to get him sleeping in his cot, we can talk about different sleep training approaches you can use. If you want to continue BFing to sleep then I would just do that for now. I think its key to get a sleep routine established first and then if you want to, we can address the independent sleeping issues.
If he still refuses to nap even at say 10 or 10.30am, then I think I would just try for the 1 nap, at around 11/11.30am. At his age, keeping him up until after lunch without him getting overtired will be a struggle. If he does settle, let him sleep as long as he wants, and then do an early bedtime. Then over a period of weeks/months, you can very gradually shift the nap later until its after lunch. My DS only started napping after 12 at around 16 months, up til then it was always before.
Oh & finally, I would try & have the CM do naps at the same time as you, so the routine is consistent.
Epic post but I hope its given you some ideas to chew over!xx