Pretty much, its unlikely the military people would even know the eruption occured- it'd be reported in a very small corner of the newspapers and it'd be big news in Iceland but elsewhere...nothing much would happen. The US occupiers would do a little investigation into how big the eruption would be and whether Iceland is safe and then conclude all is good and life will go on.
However....if we assume a much worse eruption....early tests of jet planes are smashing failures as they become clogged up. If we want to make a cool ASB scenario we could use this as a catalyst for 'WI no jets'