What happens to Gollum?
In respect to this, how did Gollum die?
The lifetime remaining to the Ring-bearer would've been brief. So the death of Gollum at the very end comes as a kind of relief from agony. He does in fact possess the Ring very briefly, just before dying, and therefore he dies having one last moment of bliss and happiness.
Similarly, what would happen if Gollum survived?
If everything happened as in the book, but Gollum survived simply because he accidentally dropped the Ring - but not himself - into the lava, then the appropriate comparison would be Frodo. Gollum would likely suffer very similar illnesses to Frodo. He would continue to age normally until he died of old age.
It was very apt that Gollum died with the ring as he probably wouldn't have survived its destruction. The reason Gollum continued to live despite the Ring being absent from his possession for so many years was because after nearly 500 years of having it, the One Ring had transformed him entirely into something else.