If you fired the NOx at sea level then you would get a power boost over and above the rated output, but it was obviously used, much in the same vein as a Super/turbo charger to give back some of the tail-off of power at altitude.

There is a view that using NOx isn’t too detrimental as the cooling effect offsets the additional heat created, but the strain on the reciprocating parts must be taken into account.

I am mystified why the ‘Compound’ engine didn’t really get developed, where the exhaust gases are passed through an impellor directly attached to the Crankshaft (a la Constellation)

Compound engines are excellent for continuous power for example an airliner or bomber, but less useful when numerous throttle movements would be necessary, as in a fighter. This is demonstrated with the US army air force bombers Vs US navy fighters. The Army wanted to use the Turbo supercharger on all its aircraft, and all the heavy bombers were thus equipped and performed very well. The compound engine was an extension of this technology, but was not sufficiently developed and therefore not available during the war and was overtaken by the turbine engine, again a further development of similar technology.

US fighters with the turbo supercharger were far less successful, until employed in roles other that that for which they were originally intended. The P38 is the best example of this, performing better at low altitude in the fighter bomber role than as a high altitude interceptor. The P47 was more successful at altitude but was again better employed as a FB. The US navy developed two stage superchargers in a parallel to the developments at RR and the Hellcat and Corsair were very successful in their intended roles, despite using the same basic engine as the P47. Horses for courses as they say.

Another point to remember is the contribution of the exhausts to top speed in a fighter. In a turbo supercharged engine some of the exhaust energy is recovered via a turbine. In a full mechanically supercharged engine in a fighter the exhaust energy was used as jet thrust. On a Merlin at full chat and at 400mph this could be in the region of 300hp. This is a lot of power to dispense with for a recovery turbine that out of design necessity had to be optimised for either cruise economy or full power