What operational differences were there in WWII for constant speed vs. fixed pitch propeller warbirds?
When war broke out, almost all high-performance military planes on all sides had variable-pitch propellers. Not necessarily constant-speed types, but adjustable nontheless. There was no difference between German and British planes.

If there was a difference it was the use of manually-adjustable propeller pitch, which was widely used in German airplanes, versus constant-speed propellers, which found more use in British airplanes. However, both sides had both types of propellers in their fleet, so it is hard to make a generalization.

Operationally, the manual adjustment gives the pilot more control, but increases workload because the pitch has to be adjusted when flight speed changes. This was less burdensome in bombers, but in fighter aircraft a manual adjustment would keep the pilot busy during dogfights.

Early types of the aircraft involved did indeed use fixed-pitch propellers, like early versions of the Me-109 A and B or the Hawker Hurricane Mk.I. By 1939, both types had long since converted to variable-pitch propellers.

Older airplanes with fixed-pitch propellers were still used for training or liaison, but not near the front lines. Exceptions are aircraft like the Polikarpov Po-2 which was used for night attacks until the Korean war. It had such a limited speed range that a variable-pitch propeller would not had made sense.