I don't think the US would have flourished, or anything, but I've got confidence in its ability to at least survive, same for Mexico. And I may just be a suffering optimist, but I think that in an apocalyptic scenario the whites, Hispanics, and Indians of the Southwest would have been able to band together to rough it out, at the very least.
Also, why would the British Empire ever relocate to India? Even here, one of its white dominions, Australia (technically not a Dominion yet, but still), is alive and kicking. Why didn't Queen Victoria just move to Sydney, or Melbourne, or more likely Brisbane since it's closer to the equator than either of those other choices? Seems like Stirling needs to think about his scenarios more.