I had WWII Fighters running on a Vista system about a year and a half ago. I'll install again in the next few days and report back here.
In the meantime, I suggest that you not install to the default Program Files folder. I think (but not sure now) that the demo is timed; at the very least, it rotates between two missions and each time you start the demo you alternate between the U.S. mission and the German mission. I have a feeling that Vista's tendency to overprotect any writable files within the Program Files folder is getting in the way of your playing the demo.
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Remove before flight