I can't really see a way to switch the two, but i don't have enough working knowledge to know how they developed. I'm thinking maybe baseball is transferred to the Isles during world war II, though how you could get the Brits to play an American game i have no clue.
Cricket i could see becoming popular in the states with live television, though i don't know that it differs enough from baseball to catch on or overtake it.
I think part of the problem comes from the complexity of the two games. I grew up playing baseball, and it was only recently i realized how boring it must be to watch if you don't know everything that's going on. Like just understanding the dynamics of how the game is played, when tension is low or high, really factors into enjoying it. Soccer is far more straightforward, basketball i figure as well. The window of opportunity is pretty small to have baseball both popular enough to export and then weakened enough to have cricket overtake it. There is also the factors of basketball and football, which have eclipsed baseball in popularity. I dont know if the OP is American, but let me tell you that there is very little room in this country for new sports, as much as i wish there were.
On a side note: Butterfly away baseball, you may get cricket in its place, though don't forget we still primary drink coffee because as a nation, we don't really like the British or British things, especially the farther back you go (other than music). To get baseball in Britain, just have the US go fascist and conquer the isles. Then you'll have the Queen at a Yankees game.
