Please clarify -- "alternate nights"? As in, Monday with mom, Tuesday with dad, Wednesday with mom, Thursday with dad, and so on? Is that right?
If so - is this a court-ordered custody arrangement? I am guessing it isn't, because I cannot imagine any family court allowing that level of back-and-forth with a child this young. Typically it would be more like weekdays with one parent and weekends with another, or one week at mom's and one week at dad's or other variations, but not one night here, one night there, with a third adult (babysitter/Grandma) in the mix too.
No wonder the boy is "hard to handle," which you note three times in different ways in your post. He is horribly, deeply confused and also not getting consistent, regular and well scheduled rest -- anywhere. The inconsistency is going to produce a difficult, temperamental child with a host of issues soon (lack of security; ability to play one parent off against the other to get what he wants; ability to get Grandma to do what he wants in order to keep the peace with checked-out, "I'm done" dad; and so much more lies ahead with this lack of consistency.... He's too young to do all that now but it will come if this keeps up.)
Mom and dad (I'm figuring mom is you?) need to work out a consistent schedule where the boy is not on an alternating night arrangement and where the boy does not have (or lack) these naps that are manipulated by the adults for their own needs. If you do not have a court ordered arrangement for custody and visitation, it's time to get one, and if dad and grandma get upset, get a mediator or a child psychologist involved -- the latter can tell them why this arrangement is not going to produce a child anyone wants to cope with. Oh, and parenting classes should be a condition of any custody and visitation arrangements for you both -- dad sounds like he's the one who really needs it but you should go as well, to show you are willing to work on the whole arrangement.