It's doubtful American soldiers would have been instructed to shoot on sight, as it was not exactly the norm to shoot unarmed political or military officials (though it certainly happened). He might have tried to escape like Goehring, but probably wouldn't have gotten very far given the extent of the Allied occupation. Had he been captured by the US, he would have been tried at Nuremberg for crimes against humanity and found guilty, then executed by hanging like most of the other high-level Nazi officials of the time. Had he been captured by the Soviets his fate would likely have been the same, though it's possible he'd have "accidentally" been executed by a Russian vigilante before then.

Hitler's death was corroborated by numerous reliable eyewitness accounts. It's not impossible, but it's about on par with conspiracy theories about a second shooter in the JFK assassination.