Renard, IIRC the HS125 used an APR system (automatic performance reserve), and this was not a matter of simply increasing thrust. Memory is failing, but AFAIR the thrust increase after an after an engine failure was to achieve the value necessary to meet the minimum performance; thus it is a necessary increase.
There is an AC / Regulation on this somewhere (I’m looking), but I note that aircraft which use such systems have low thrust asymmetry with engine failure, i.e. centre-line engines.