We can actually use Chewbacca's stated age in the movie, compared to his stated age at the end of the Clone Wars, to determine fairly accurately when Solo: A Star Wars Story takes place.
Within the movie, Chewie tells Han his age.
Han Solo: 190 years old?
[Chewie howls again]
Han Solo: You look great!
Solo: A Star Wars Story (2018) - Quotes - IMDB
Based on this answer from Wad Cheber to the question "How old is Chewbacca?", we get a Disney canon
answer to Chewbacca's age at the end of the Clone Wars.
He was a Wookiee, after all, and he knew passion. His were a proud people, a people who had lived for hundreds of years peacefully on their wooded homeworld of Kashyyyk until the Clone Wars. He had been younger then, just one hundred and eighty, and he had fought the Separatist battle droids. He had witnessed the betrayal of the clones and the beginning of the Empire. He had seen his people, his brothers and sisters, his family, put in chains and sold as slaves throughout the galaxy. He had been put in chains himself, and just the memory of it made a growl rise in his throat.
Star Wars: Journey to The Force Awakens: Smuggler's Run - A Han Solo & Chewbacca Adventure
Since the Battle of Kashyyyk was fought in 19 BBY (which we see during Star Wars: Episode III), and Chewie has aged approximately ten years in the process, we can reasonably determine that the movie starts in 12 BBY, then jumps forward three years for the rest of the movie to take place in 9 BBY. This places it after Episode III, before Rogue One and Episode IV, and before Star Wars: Rebels.