A general point here that hasn't been made so far. Every approach should be made with the expectation that you will NOT be able to land off it and that you may have to go-around. To that end the Missed Approach Procedure (MAP) should form part of your approach briefing, and should be covered considering both the lateral and vertical profiles, and also how you actually fly your aircraft.
To return to your example: If the needles have gone full scale deflection, and the ILS is still working then NO you should not continue on your heading. It was this heading that has gotten you off the ILS in first place! Depending on how far out you are you should try to recover the ILS. If in doubt go around doing your best to recover the published MAP.
In your example of steep valleys. I presume you are talking of a hypothetical situation, and I can only echo the advice given by others here, brief thoroughly, and fly accurately. If the ILS itself fails then go-around advise ATC and ask for any assistance they can give you via radar.