There's been a lot of commonsense replies so far and I shall attempt to provide a bit more.
First, I'm not even going to contemplate how any pilot could allow such a sustained tracking error that FSD on the LLZ would be achieved without some sort of corrective action - heading change in the early stages, while it's still viable, or an immediate go-around before it becomes unviable. Any pilot who can allow such a situation to develop, whether manually or by ignoring a possible foul-up in the automatics, is not the sort of pilot who should be in a cockpit.
Second, while I can't imagine such a situation, if the LLZ suddenly went to FSD, I'd already have a fair idea of how I was tracking immediately prior to that event. If I was on the LLZ, it's a fair bet that my heading was appropriate so I'd stick with that while initiating the G/A and climbing for all that my aeroplane is worth. If I wasn't actually on the LLZ, at least I'd be within my tracking tolerance and, depending on circumstances, might make a slight, strategic heading change while initiating the G/A and, once again, climbing at the maximum safe limit.
If your next question is... how do I identify the point where the missed approach should start? Well that's a good question and the way in which a pilot handles it is part of the reason why you get paid the big bucks! Of course, the DA is the point of commencement of the missed approach, so you need to think about the ways in which you might identify it. Sometimes, there'll be an Inner Marker, or a published DME distance - or at least a way to identify that DME distance. As my aircraft is equipped with a TSO C-129a GPS, I'd already have the RWY THR position selected on it, if it's in the database. If all else fails... well, what can I say? You need to exercise your own best guess as to the location of the point that equates to DA, as a distance from the THR. There again, you're being paid to exercise command judgement that is intended to keep your pax safe.