Interesting points BOAC. You're right that getting down the ILS in solid Cat III is not a problem but from the controller's perspective getting the aircraft onto the ILS safely would be my concern at most airports. Given that RADALT only works at low levels - perhaps below the normal safe intercept level - and airports tend to be in 'terrain rich' environments, from a controlling perspective it could be somewhat challenging.
As you say, in some conditions it will be a major emergency for the crew, but in others less so. As a controller, if I were faced with this situation I would do everything I can to ensure that the flight can be continued without the crew needing to be concerned about maintaining a level accurately. A combination of VSI and mode C (I'm assuming no major failure of an ADC in a sophisticated aircraft here) should be good enough if the levels above and below the problem aircraft are kept clear. If it's a major emergency I presume that the crew would be looking for a nearby suitable airport at which to land so continuing the flight and navigating become less of an issue.
But as I say, some interesting discussion points. And everything that I've offered is based on a well supported ATC system - because the question was about ATC procedures. For a pilot flying over some parts of the world, I imagine there would be a whole range of additional considerations!