Try to determine ownership. It's not clear whether your great-uncle had a will (and the copyright wasn't mentioned in it) or whether he didn't have a will. If he had a will, and the copyright wasn't mentioned, it would likely go to whoever received the residuary estate. If he didn’t have a will, you would need to map out the family tree and determine what blood relatives were alive at the time your uncle died. For example, if your uncle had no will and your grandmother was his closest living blood relative at the time he died she probably became copyright owner under the rules of intestate succession (you can see how it works if he died in California). So, initially the copyright went to whoever received the residuary estate or by intestate succession. If it went to your grandmother, then, upon her death, the copyright would have passed according to her will, or if she didn't have a will, intestate succession would kick in again.
The diaries and the copyright. Your grandmother may have given you the diaries as a gift but giving you the physical possession of the diaries is different than giving you the copyright. It's the difference between giving someone a painting, and giving someone the right to reproduce the painting. That's why copyright transfers must be proven by a written document. By the way, the unpublished diaries are protected under copyright for seventy years from your uncle's death.
If this seems too confusing ... If this seems overwhelming, keep in mind that what you’re really trying to determine is whether there is a relative who will object to your claim of ownership or seek a share of any revenue from the publication. So, if you don’t expect the publication to be a major source of revenue, it’s likely you can proceed without bothering with most of these ownership rules because battles usually only arise when a big bankroll is at stake. If the diaries generate revenue and you're still unsure about ownership, put the money in an account for a period of several years in case someone appears out of the woodwork to challenge your claim. That good faith effort will likely go a long way towards resolving the matter.
Copyright applications. By the way, if you should file a copyright application, you would list your uncle as author and yourself as copyright claimant and under “transfer statement,” you would choose "By Inheritance" (as shown above). Also, if you should file a copyright application, you will have to make a statement under oath that the information provided about ownership is correct.