hi there, interesting subject. Well despite the fact that auto land doesn't have anything to do with weather it very much depends on the quality of the ILS signals and accuracy. A little background information of two ILS facilities ILS CAT I does have two waves 150 and 90 Hz both on LOC and GS whereas CAT II-III ILS has got additionally a carrier wave which enhances the accuracy significantly. Here are the parameter of both ILS certifications
CAT I max deviation LOC +/- 10m TDZ; GS +/-3m TDZ
CAT II/III max deviation LOC +/- 5m TDZ, MID, ROLL OUT, GS +/- 1,2m TDZ
As one can see there's quite significant difference in accuracy and area of application ergo the recommendation. Theoretically one shouldn't ever experience a LOC shift on CAT III ILS ground facility.
Auto land is not CAT 3. An automatic landing system is only an equipment providing automatic control of the aircraft during the approach and landing and is not related to particular weather conditions. This system is mandatory for all CAT 3 operations. However, it is a common practice to perform automatic landing in good visibility but in that case, the ILS performance must be sufficient and ILS signals protected.