There is a lot of debate about whether The Lord of the Rings: Gollum can be considered canon, which is normally the debate for any game or movie made about the series. The company Middle-earth Enterprises, which owns the motion pictures, merchandise, stage, services, and other rights to The Lord of the Rings and The Hobbit books, has worked closely with the game, according to a recently released Steam FAQ.
How closely does The Lord of the Rings: Gollum follow the lore?
You control Gollum, so not every action you take as the character is canon, but all the events and characters are. Developer Daedalic Entertainment clarified whenever a new piece of material is "officially accepted into the storyline," it is canon. It is confusing, but basically, if it is mentioned in the books, it is canon, but if it is something entirely new, it needed to be approved by Middle-earth Enterprises.
Middle-earth Enterprises reviews all elements of the game to ensure that they follow the story as closely as possible. This is as close to canon as you can normally get. If all you consider is the works of J.R.R. Tolkien and his son Christopher Tolkien, then no, none of this is canon. J.R.R. Tolkien died in 1973, and Christopher Tolkien died in 2020, so nothing made after that will be canon by those standards.
When does The Lord of the Rings: Gollum take place in the lore?
The Lord of the Rings: Gollum takes place in the early chapters of The Lord of the Rings: Fellowship of the Rings book. This would be around the time that Frodo decides to take a shortcut across the fields between Woody End and the Brandywine River ferry. Frodo is about to embark on his journey at this point, so it makes sense that this will be the time when all of this takes place.
For more on Lord of the Rings, check out 4 things we want to see in a Lord of the Rings mobile RPG or All Mobile Lord of the Rings Games, Ranked here on Pro Game Guides!