When did the military/Air Force start measuring fuel in pounds instead of gallons?
Good question! It makes sense for the conversion to have been done in 1951, as the film you saw states, because that is the year when JP-4 fuel was introduced. JP-4 has a lower freezing point (-60C) than earlier JP-x fuels, which were all derivatives of JP-1, first introduced in 1944. Lower freezing point means the aircraft can fly higher, and the changes in volume would have led to the change in measurement basis.

The reason is that the weight of fuel doesn't change with temperature or pressure, but the volume does. Rule of thumb is about one percent change in volume for every ten degrees centigrade up or down.

A little experiment to prove this would be to weigh a glass of water, cool it in the freezer for a few hours, and weigh it again. 

If fuel quantity indicators on jet aircraft were based on volume they would indicate different levels depending on the aircraft's altitude and outside ambient temperature. That would not be very useful.

It does also make weight and balance calculations easier, otherwise the loadmaster would have to compensate for temperature when calculating center-of-gravity.