Why did a former emulsion of oil and water precipitate after heating and cooling when heat made it originally?
The simple explanation is that the speed of de-emulsification depends strongly on physical state and temperature.

If you have created a solid emulsion by cooling the initial blend, the de-emulsification will be very slow as there is little scope for the components to move around at all never mind separate. warming this solid up so it liquefies creates two factors that make separation easier: the mixture liquefies and the temperature rises, creating many easier ways for separation to occur. Once separation has occurred only re-blending will restore the emulsion.

Emulsions often separate slowly (milk, for example, slowly forms a layer of creamier fat on top but it doesn't happen quickly after thorough mixing). In your case the initial cooling "froze" the emulsion but the microwave heating enabled de-emulsification to occur by heating the mix. Cooling the result without blending is not going to restore the emulsion without further blending.