Why did most WWII Allied piston engines not use fuel injection?
Those engines didn't use carburetors as in gravity operated float carburetors.  They used "pressure carburetors".  Pressure carbs are mechanical fuel injection systems that inject the fuel from a single point at the start of the manifold instead of injecting fuel at the intake port via a distribution system.  Not quite as theoretically efficient as point fuel injection, but it has all the other advantages including automatic mixture control and no G sensitivity, as well as being less sensitive to damage by not having small high pressure fuel lines running to each cylinder.

On cars in recent years you had a similar system called "throttle body fuel injection", more or less an electronically controlled pressure carb.