The rectifier tube shouldn’t make any difference in sound.
’Shouldn’t’ is why this statement is true; in practice the rectifier will affect the sound. This is because the rectifier has a voltage drop across it, which varies according to current draw and from tube to tube (and rectifier tube type as well).
Amps designed with a tube rectifier will certainly sound different if a different rectifier is used simply because the B+ Voltage might be changed- if going from tube to solid state the change could be as much as 75 Volts in the power supply (which might challenge existing power transformers, filter caps and the like, so don’t do it...).
If the amp were to employ enough feedback it could be able to reject the effects of the rectifier, but no tube amp made can employ the amount of feedback required! So in practice, rectifiers affect the sound, plain and simple.
Power tubes and driver tubes probably affect the sound more though...