What would happen if I made angel food cake with whipped cream instead of whipped egg whites?
It would not work. The benefit of egg is that it is largely (aside from water) comprised of protein (~12g/100g). This forms a stable structure when you whip it, that is maintained because the proteins denature (that's why the white goes white from clear when cooked) and form an insoluble and highly stable mass. 

Cream, on the other hand is largely fat (19g/100g), which forms an emulsion when whipped. When you heat fat - it melts. This would mean that the batter would lose its air and structure.