He seemed to give the Straw Hat to Shanks pretty early in his career, seeing as how Shanks wore it during several confrontations with WB and had it a fair amount of time before the battle of Edd War, so I don't think Roger's decision would change any more when it seemed he didn't become aware of the illness until after the battle with Shiki.
I think Roger would have definitely remained the Pirate King. I doubt any of the Yonko would team up to take him on when considering Whitebeard would likely side with Roger in such a conflict and Shanks would still likely make the same progression to a Yonko as in the main timeline. Considering Roger has been painted as being a more ruthless version of Luffy, I doubt he'd limit his crew's desire to pursue their individual dreams, so Shanks would still likely travel around, meet Luffy and trigger the events of the story's beginning.
The main difference though is that without Roger's death or declaration of the One Piece, the Great Pirate Era likely wouldn't begin and most pirates would likely instead be aspiring to overthrow Roger. Depending on what's later revealed regarding Roger's knowledge of the true history and what he aspired to do with it, the World Government may very well have been overthrown by the time of the main story if he chose to make his discoveries public. That could change practically everything, though I wouldn't be able to make a concise scenario until the series actually ends.
