I don’t think it’s really plausible that any of the three major Axis powers would have been able to launch an invasion of mainland America large enough to actuallyhewould not have gained ground against the Americans fighting on their own turf. While the German army may have matched that of the United States, their navy was in no way ready to launch any amphibious invasion, especially one of the scale requited to attack the US. Italy… let’s just say I couldn’t see Mussolini walking into the white house.
Most likely, if the Axis were able to overhwelm all of the other allies, a cold war would develop between the US and Germany/Japan (and maybe even between the axis themselves) and all sides would race to develop defenses and nuclear weapons.
If any axis takeover of washington was to take place, it would be a result of either a revolution within the US or (less likely) an invasion by Mexico and other Central/South American countries supported by the Axis. Germany would likely send another Zimmerman letter, but this time would provide actual support. Although by this point, with the US in possession of nuclear weapons, i doubt and invasion would take place.