In the US, the FAA will not approve an Operations Specification for CAT III operations (Ops Spec C060) unless a crosswind limit of 15 knots (including gusts) or less is specified. This is also true for CAT II; however, the CAT III number used to be 10 knots, and was changed to 15 in recent years. I do not know the reason for these limits, but I suspect it has to do with the "seeing condition" requirements that also drive the minimum RVR for those approaches. Neither of these numbers has anything to do with the airplane's capability; the B757 is limited to 25 knots of crosswind during an autolanding, as well as 25 knots of headwind. I suspect that the airplane limits are driven more by signal gain issues, such as the autoflight system's ability to identify tracking trends due to gusts in a timely manner. Control authority, as far as I know, is not a problem.
One point that is often overlooked is that, from the FAA's perspective, the crosswind limit for CAT II or III must be observed at the time of touchdown. It is not subject to the same type of "approach ban" criteria as minimum RVR. You have to be within limits at touchdown, no matter what. So theoretically, a tower report of RVR below mins once inside the final approach point (1000 feet for UK) would not force you to abandon the approach, whereas a tower report of crosswind in excess of limits would.