That's the big question, why did it take so long to develop the mechanical two stage supercharger? Started in 1940, it finally made production in April '43. It's just a simple impeller in a diffuser driven from the engine by a jackshaft. Three full years to develop what should consist of off-the-shelf parts. High altitude performance without all the ducting of the turbo (14' per engine on the P-38 Lightning) and it didn't need an intercooler except for WEP. If you can figure out why this took so long to develop please tell me.
A turbo in the rear fuselage (instead of under the engine) like the P-47 would have caused (more) balance problems.
Oh boy , oh boy, oh boy.
Allison first toyed with idea of a 2 stage supercharger in 1938 or 1939 but quickly gave up as they didn't have enough engineering staff to work on it plus all the "stuff" the army wanted at the same time (Like pusher engines for the Airacuda).
first example (or more than the first) did use "off the shelf parts". The tried using the same impeller as the engine supercharger, I don't know about the diffuser. They soon learned that far from the optimal set up, very far.
When dealing with these superchargers you have several things going on. What is wanted for making power in the engine is mass flow of air, that is to say pounds per minute. But that is hard to measure but pressure isn't. So they use pressure gauges but they are only indicators. Since the weight of the air per cubic foot varies with the temperature even at the same pressure they have to make some assumptions in aircraft (they have a lot more instruments on a test bench/test cell). ANd as you go higher the density of air changes, by the time you get to 20,000ft air is just over 1/2 the weight it is at sea level per cubic ft. You need a much bigger intake system (inlet duct/s and supercharger inlet) to handle the low density air. Once you have used a 1st stage supercharger to squeeze the air down to something like sea level air your 2nd stage impeller-diffuser can be about what it was for the single stage engine.
RR got "lucky" in that they had a large supercharger already "in house". They just grabbed the impeller (for sure) and housing (maybe) from the Vulture engine as it flowed about the mass of air they wanted for for the hi 20,000 ft to 30,000ft area. This was for the first test rig, later prototypes and production engines used purpose designed impellers and diffusers.
Allison had no such large supercharger or impellers hanging around and no good place to get them.
You need intercoolers to make high power at altitude. Otherwise you are just adding weight, bulk and cost to make a slightly better
low altitude engine. The supercharger adds considerable heat to the intake charge. For instance the supercharger on a Merlin XX added 148 degrees celsius (over 260 degrees F) to the temperature of the incoming air when in high gear. using more compression in the supercharger/s just adds more heat and at some point you reach the detonation point.
The higher temperature if using higher 'boost' without cooling means the air is less dense which means you get less power from the same 15lb of boost than if the air was "cooler" and at 15lbs boost. The heat load on the entire engine goes up. Raise the heat of the intake charge 100 degrees and peak temperature in the cylinder goes up 100 degrees and the heat of the exhaust goes up 100 degrees.
And you wind up having to use lower intake pressure because of the higher temperatures. Say your engine can handle 15lb of boost with an intake temperature of 300 degrees (hypothetical figures) with an intercooler. take out the intercooler and lets say the temperature goes to 400 degrees. Now lets say that extra 100 degrees pushes the mixture to a point where, as the piston rises and further compresses the mixture, it raises the temperature to the flash point and the mixture either ignites on it's own before the spark plugs fire or the when the spark plugs fire the added pressure of the flame front cause all the mixture to explode at once rather than burn in an advancing flame front across the cylinder as the cylinder goes past top dead center and starts to descend.
While using two stages gets you less temperature rise than using one stage for a given level of compression trying for high boost (and a Merlin 61 was compressing the air over 5 times in the superchargers at altitude) without using an intercooler runs into diminishing returns real quick.