My understanding is that the second Children of Ilúvatar are called to Mandos when they die and then move ... elsewhere. I think this applies to all men, halflings, etc and the only way to avoid it is not to die (which Gollum managed for a while). Also I think all second children move on, unlike the elves who can be left stranded in Mandos if they were evil, though I don't know if moving on involves any sorting into different destinations such as Heaven or Hell.
So I guess the obvious answer is that yes Gollum did go to Mandos, then elsewhere to some unknown fate since he was originally a hobbit, but do we have any evidence for or against this in the books or statements from Tolkien? It isn't obvious to me that there was enough of Sméagol left in Gollum to qualify him as a man.