There is real world, and there is checkride. Real world, unless I'm very low on fuel, I'm going missed, get clearance for another approach, rebrief for the new approach, and then fly it. For a checkride, or if I'm really worried about fuel (flying a 412), then I may continue, depending on the weather, my situational awareness, my confidence in my SIC, and many other factors. It's certainly legal to continue, using the localizer minima, and I don't think that is spelled out anywhere. Whatever is not prohibited is legal, at least in the US, and as long as you have a clearance for an approach and don't descend below the minimum altitude for the approach, and can fix the MAP properly, then you're legal to fly the approach. Legal and safe are not necessarily the same, however, so I would prefer to rebrief and start the approach over. I don't think it's possible for the examiner to fail the applicant for doing this, nor do I think he should be failed for continuing, assuming no minima were broken.