This one's going to be a bit type-specific.
Like our fellow Prooner
Wizofoz, I too fly the B777 with "Auto Rudder" (OK, yeah, I know that Boeing call it TAC, a cr@ppy term), Auto Throttle, and no Flap configuration required to continue the approach, there would be negligible (if any) deviation in Localiser or Glide Slope, so, as he says..... keep on truckin!
If there's no "Auto Rudder" and / or a Flap configuration is necessary, yaw and/or sink is very likely and manual throttle, ILS deviation could become necessary as
BOAC has pointed out, so a Go-Around might be very desirable.
For a jet, no engine "securing" is an urgent matter, leave it alone until on the ground. For a Prop aircraft, feathering actions etc., might again necesitate a G/A/
It's all type-specific, and in accordance with company SOPs.
Regards,
Old Smokey