Had the Browns made the move in 1942, as the article speculates, I'd guess either the A's or Senators would have jumped to SF within a very few years.
With the AL teams in California drawing like gang busters for the first few years, the NL would probably have jumped to fill the next massive major state still without MLB with expansion teams in Texas. I'd guess expansion would have come in the early mid 50s.
The Upper Midwest cities of Minneapolis, Milwaukee, KC and Denver would be wide open for moves too.
Of course, had Pearl Harbor never happened, WWII would have unfolded entirely differently, on a different timeline, and with significantly different outcomes in many ways -- not necessarily High Castle, but the economy of the nation might have been entirely different.
Say, for instance, instead of Pearl Harbor pulling the US into the war with both feet, the US enters into both the European and Pacific wars over the course of several years; there's no instantaneous explosive ramp-up of the war manufacturing machine, instead things drag on and on. Say the war lasts from 1944-1952. So then when it gets over in 1952 or whenever, there isn't the massive manufacturing economy ready to take off.
Sending baseball expansion teams to Atlanta or Seattle probably wouldn't have been very high on the priority list.