"Going native" in this context means sympathizing with the local population to the extent that their loyalty effectively shifts from the United States to the local people.

I suspect the Americans have this problem due to cultural factors. The binding ethos of the United States includes a hefty (if not total) tolerance to the idea of nationality being less important than their ideals.

The chief unifying principle of Great Britain is ultimately old-fashioned nationalism. They have an expectation that other peoples should become more British, rather than that they should accommodate new cultural norms.