Did WW2-era aircraft have constant-speed or variable-pitch props?
Variable pitch (with feathering) was virtually universal on (at least American) multi-engine fighters, bombers, and transport aircraft even early in the war: the DC-2 had feathering variable pitch, and it entered service in 1934.  The P-38 had feathering and variable pitch, and was in service before the US entered the war.

Constant speed propellers were also found on many single engine combat aircraft -- one was the Hellcat, which entered service in 1943, though it's very possible the technology was introduced earlier.  The P-47 Thunderbolt also had this feature.

When you're talking about aircraft that weigh several tons and need to deliver thousands of horsepower through a propeller, especially for combat operations, the weight and cost of variable or constant-speed propellers became negligible by comparison to the cost of the aircraft or the cost of fielding a losing design.