Most x-wind limits in large jets are max "demonstrated". This means that was the highest wind the manufacturer has tested it in. You aren't limited to that wind speed, but if you bend something while landing in something higher than those winds the feds are going to slap a "careless and reckless" on you. If you don't bend anything, nobody cares. I usually use the max demonstrated as my max also. We also have a max x-wind limit of 20 kts on wet runways on the B767.
About Cat II/III landings. Most modern aircraft are capable of autolanding and usually the airline's ops specs/AOM requires the crew to autoland in anything less than Cat I mins. Some carriers can hand fly down to Cat IIIa mins (50' DH & 700 RVR) using a HUD like our B727's. Never used one so I don't know what their x-wind limits are and someone else will have to answer.
For UPS B757/767 aircraft the x-wind limit for autolanding in Cat II/III wx was 10 kts. The aircraft autoland limit was 25 kts but retrictions apply during very low vis landings.
Realistically speaking, it's a mute point since most Cat II/III landings are due to fog (unless your landing in a severe T'storm with heavy rain...and why would you being doing that?) and you're not going to have much wind in foggy conditions.