Wizofoz:
At no point. You can continue to land after an engine failure at any stage of the approach, and it is in fact preferable to now doing an anannounced, unexpected single-engine go-around.
ATC has to be ready for "unexpected" missed approaches, especially when the weather is at minimum CAT I conditions as set forth in the OP's hypothetical.
First and foremost, like BOAC states, the LOC and GS better continued to be "nailed" both during and after the engine failure. If, on the other hand, the pilot is now chasing an unstable approach while continuing to descend on the G/S he is creating a far unsafer condition than a one-engine inoperative missed approach.