I'm probably missing something, and the Stones probably affect the timeline much earlier, but this is my (not very exciting) what-if scenario.
Captain America: The First Avenger actually might not be terribly different. What If? claims that HYDRA is still a threat without the Tesseract, and its nonexistence shouldn't affect Erskine developing the Super Soldier Serum. So Captain America still defeats HYDRA, but maybe he doesn't get frozen.
Captain Marvel doesn't happen because Mar-Vell doesn't stay on Earth. No Tesseract there, so she'll look for other power sources to save the Skrulls. The Incredible Hulk probably happens the same way if Erskine's unaffected.
Iron Man is screwed. Even if Stark Industries doesn't need the Arc Reactor (which Howard Stark and Ivan Vanko invented based on the Tesseract) to do well, Tony will still get kidnapped by the Ten Rings, and without the Arc Reactor, probably never escape.
Thor either happens similarly or not at all. We know the Asgardians fixed the Rainbow Bridge between The Avengers and Thor: The Dark World using the Tesseract. If they needed the Tesseract to build the Bridge in the first place, that could affect Odin's rule or even his conquering of the Nine Realms to begin with. So if they didn't need the Tesseract, things go similarly. If they did, Odin might still be bloodthirsty or have been defeated, Hela might not have gotten banished, and Earth has no Norse mythology.
Thanos stopped massacring civilizations personally because he knew the universe was too large for him to manage by himself. This led him to seek the Infinity Stones, so either 1) he kept massacring them personally, 2) he sought something other than Stones, or 3) he gave up. Either way, The Avengers doesn't happen, especially if Loki never fell into Sanctuary. Natasha and Clint still work for S.H.I.E.L.D., while Bruce is still in India. Tony and Steve are probably dead. Thor might not exist.
The Guardians of the Galaxy don't form because there's no Power Stone. Ant-Man is probably mostly unaffected, unless Hank Pym used the Tesseract to make Pym Particles or something. Doctor Strange is problematic. Either the Masters of the Mystic Arts have something other than the Time Stone to battle Dormammu, or the world ends. Spider-Man might still exist, since his origin is pretty vague. Probably stays low-level, though.
I don't think the Wakandan meteorite is affected, but Zemo never killed T'Chaka, so Black Panther is different, with Killmonger challenging T'Chaka's rule instead of T'Challa's. Either T'Chaka denies the challenge or accepts it, at which point T'Challa might then challenge Killmonger should the elderly king lose.
So by the end of Phase Three, we have a dead Captain America, a dead Tony Stark, a Hulk in hiding, Black Widow and Hawkeye still working for S.H.I.E.L.D., Ravager Peter Quill, Ant-Man, Spider-Man, and Black Panther. There's also no Norse mythology and now that I think about it, Ego probably consumes almost everything in 2014, promptly followed by Dormammu consuming everything.
