German supercharger technology was behind that of the Allies throughout the war. RR knew more about building efficient superchargers than anyone in the world, though P&W wasn't far behind. The other major disadvantage the Germans had was inferior gasoline. They started the war with 87 octane. By '42 it was a bit better, but by war's end they were back to 87 octane synthetic gas derived from coal tar. The Allies also began the war with 87 octane. By the time of the Battle of Britain, the Brits were using 100 octane obtained from the U.S.A..
By 1944, the Allies were using 150 grade fuel and were experimenting with 170 grade on a limited basis. An engine's power output is in direct proportion to the octane level of the fuel. Boost octane and you can boost the power, assuming the engine is strong enough to handle it. German aero-engines were able to keep pace (more or less) with the Allied engines by having notably larger displacements than their rivals. The Ta-152H, the ultimate development of the FW-190 series, used water/alcohol injection and liquid nitrous oxide injection to boost power at high altitudes. Yet another clever solution that came far too late to be of any real help.
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What you say makes a whole lot of sense with the RR putting out super efficient superchargers. I know the mustang was suppose to be no good above 15,000 feet or so until it was fit with the merlin engine. I think the british were pretty quick to dump the american engines out of their p40's too. I guess when you think about the only high performance fighter that was all american in design was the corsair. The british also started us off by giving us their first jet engine designs, by the way supposedly british designed engines made their way to the mig 15. They came up with radar, the first electronic computer, and the steam catapult for carriers, not to shabby.