In the States, the ILS critical areas are protected to two levels. The first is in effect when the either the ceiling is below 800 feet OR the visibility is below 2 miles. In this case, ground vehicles are limited, CAT II/III hold lines are used, etc.
The second level is in effect when the ceiling is below 200 feet OR the visibility is below RVR 2000. In this case, the preceding aircraft must be clear of the ILS critical area before the following aircraft reaches the middle marker (or, in any event, where the middle marker used to be). This is specifically intended to prevent ILS signal interference from the preceding landing aircraft at a rather inopportune moment.
If the ILS approach that Shore Guy mentions was tight, it is likely that the ILS critical areas were fully protected. This is not quite the same as LVPs in Europe, but achieves the same goal with regard to ILS signal integrity. (Harmonization note: the ILS critical area in the US is parallel with the ILS sensitive area in the UK.) The controller's remark seems to me a rather rude but pointed reminder that he, the controller, doesn't tell you what is legal for your airplane and certificate and what is not.
Having said all that, I discovered a couple of years ago that the local government versus federal government battle in the United States extends right to our doorstep. At JFK, I am told that the Port Authority controls the ILS critical area protection, and they automatically initiate the procedures of removing ground vehicles and so forth when the weather crosses the appropriate threshold. The tower separates aircraft for the requisite spacing, but does not truly "control" the surface operations in this respect.
In terms of whether you can fly a CAT III approach when the tower says CAT II ops are in use...in theory, a ground equipment degradation should be either NOTAM'ed or included in the ATIS broadcast. For example, the ATIS at Stansted occasionally appends a comment at the end of the broadcast: ILS radiating Category I" which I suspect indicates the system is not fully monitored or some such deficiency. Absent any such advisory or NOTAM, and presuming that the runway is suitably certified and the aircraft and crew are also, there should be no reason not to execute a CAT III approach.
But as someone has said, the simpler solution is to simply ask whether the airport can currently accept a CAT III approach.