Austria was known for being "The Third Reich's first victim" and in OTL, Hungary was invaded by the Germans when Horty signed a peace with the Soviets. So yeah.
To be honest, Otto von Habsburg would probably be able to spur up some resistance in Austria-Hungary, which would be a great help for the Brits fighting the Axis+USSR without the US.
To be honest, Austria-Hungary could be re-created democratically. That's something the Brits can't really stop.