Technical issues associated with adding a Cat III capability to existing ME aircraft are considerable, as would be the associated training burden. 'A few' simulator exercises would add to the total training cost and maintaining the associated pilot recency would be far from straightforward.
Whilst the VC10 might have flown to Cat III limits with ba, current regulatory requirements are considerably more demanding. I don't know whether there's been much improvement in recent years, but the reliability of an auto-coupled approach in an RAF VC10 wasn't particularly good even 7 years ago. Random disconnects at GS intercept were infuriatingly common and autothrottle speed maintenance wasn't particularly accurate.
Cat II limits offer little more than Cat I; the number of days when the cloudbase is between 100 and 200 ft a.a.l. being pretty few. Whereas Cat III, for those who need such a capability on a daily basis, offers true 'no DH' operation. At a cost - which is probably acceptable if your airline revenue stream requires it and you operate a fleet of modern aircraft.
Some years ago, the northern parallel road was relocated at Brize at considerable expense, the intention being to prepare for LVPs for the TriStar. After the work had been completed, it was discovered that the terrain even at Brize didn't allow for anything better than a Cat I ILS installation on RW26, the almost fatal crash of a TriStar on RW08 led to auto-coupled approaches even to Cat I limits being banned on RW08 - most people who have a lot of experience of flying at Brize will be familiar with the undulating glideslope on RW08!
With regard to MPA, terrain relief on the approach to the RW28 is rather less benign than the terrain relief at Brize. Whether it would even be possible to flatten the landscape sufficiently to accommodate a Cat III ILS I don't know - but the cost of the occasional rare diversion will undoubtedly be far less.
But why bother unless the Typhoon also has a 0 ft DH autoland capability?