To the best of my knowledge, none were purpose converted by the military for the military, which during the service use of the MH/ME system used a sub-caliber training device fitted to the "as issued" service arm in the form of the Morris tube, chambering the .297/230 Morris centerfire cartridge. That isn't to say that no military unit ever used some, only that the conversions were not intended for the military, but for the Society of Miniature Rifle Clubs ( http://www.rifleman.org.uk/Society_o...ifle_Clubs.htm ). The formation of the Society and the resulting mass conversion of obsolete Martini-Henry, Martini-Metford and Martini-Enfield rifles and carbines to .22 LR for their use had its roots in the perceived mediocre performance of the Brit military in the Boer war and the desire to improve military marksmanship through fostering widespread civilian marksmanship practice. The conversions were performed by multiple firms starting shortly after 1900 and continued as required to meet demand up into the 1930s. Most were converted by Bonehill or Greener, with other manufacturers being less common.
In general the same rules in collecting other types of arms apply here, but with a few caveats directly pertaining to the unique birth and use of these conversions. Since these arms were the end use of a weapon system that had seen many conversions while in service, these rifles can show some unique combinations of manufacturers and history of use prior to their conversion for civilian use by the Society. Many if not most show further modifications/personalization in the form of various sling attachments, sights and other "improvements" to the rifle through the years of civilian use. How a collector views the wide assortment of possible combinations and "after market" modifications depends greatly on their personal goals in collecting. Desirability is somewhat in the eye of the beholder and not necessarily reflected in market pricing.
An example of what can be found is my first Society conversion. It started life as an Enfield 1873 dated MH MK I rifle, then converted to MK II standards, then subsequently converted to a MH Artillery Carbine MK II, then converted to a ME rifle in .303....all before being converted by Bonehill to a Society rifle with a new barrel. To me it is highly desirable from the standpoint of the history of its various conversions. But Bonehill being a common conversion company, it would be less so to someone chasing rifles/carbines from less common converters. Similarly some collectors are focused on firearms originally manufactured by a given company, such as LSA and are indifferent as to who did the final conversion.
As to rifle vs. carbine, the carbines are less common. However, these should all be treated as parts rifles where you find carbine receivers used on rifles and rifle receivers used in making carbines. Some will have relined bores, some will have new barrels. Some that never were converted to ME rifles will be assembled using ME wood and fittings. The combinations aren't quite infinite, but are considerable. Similarly what rear sights the rifle may or may not have might be important to some and totally irrelevant to another.
Prices are primarily guided by rifle vs. carbine and the overall condition and originality of a given arm. The vast majority of sales show little price differentiation for many of the possible combinations of mark, manufacturer or who did the conversion. That said there are two examples I've seen and would have dearly loved to have acquired. The first was a MK I that had escaped upgrades/conversions until Greener converted it to a Society rifle, still retaining its bronze block pin. The other was a Bonehill conversion of a EM rifle converted to a MK IV Pattern "A" that utilized the original barrel with a liner. Both fall into the category of being truly rare (the former being almost unimaginably so) and both going for prices that reflect more than a few folks noticed what I did. Records at the level of specificity needed to comment on the rarity of a given combination just don't exist. However it is safe to say any Society marked rifle by other than Greener or Bonehill is worthy of notice.
Both Greener and Bonehill (and I'm sure others) offered similar .22lr conversions to the general public for target and sporting use. These do not bear the Society's markings and are treated somewhat like the civilian "volunteer" patterns of the MH and ME rifles sold on the civilian market. While some interesting combinations can be found there as well, the ones specifically marked for the Society support the most collector interest.