You may well be aware, as I've mentioned the fact in other threads and places, that I've first read Tolkien in Russian. And though the Hobbit translation (by Nina Rakhmanova) had distinctive taste of 'Englishness' about it, the case with LoTR was different - Muraviev and Kistakovsky (best, in my opinion, despite numerous blunders, Russian translation) obviously tried, and succeded in a way, to make LoTR 'russian' cultural phenomenon. The language is very 'living', for lack of better term, and at times it bears similarity to Russian 'bilina's (thing approximately correlative with Svandinavian 'saga's)
I ascribe to this very translation (it being the first as well) the popularity Tolkien has in Russia. It subsided of late, but there certainly was a phenomenon known as 'Tolkien craze', which burst out (and was mainly manned with young people - teenagers main part, but up to people in their 'late tweens') by the end of 80s, following publication of 'short version' translation of LoTR in 1982 and first full translation in 1988. There was much fuss about it in Russian media in 90's, there was even an insulting term coined (tolkienutiy - roughly, 'banged on the head with Tolkien'), but it was (and to a point, still is) wide and vast movement of RPG players and people involved in arts (poetry, music, costumes etc). Media accused participants of Tolkien movement of being anything from escapists (which they, up to a point, were - given intensity of social changes and collapse of the whole governmental system and istability in all post Soviet Union space in the 90s) to sword-armed maniacs and satanists. Orthodox Church was also divided in evaluating of the phenomenon - those priests who, presumably, only gathered some bits of media fuss accussed the movement of betraying 'orthodox values' for the sake of 'faulty' or even 'heretical' 'Western values', but there were actual priests in the movement itself.
Some occultists and 'would-be-extrasensors' and 'wizards' (number of which grew like mushrooms after the rain in aftermath of collapse of mainly materialistic state) tried to exploit Tolkien as 'visionary' giving glimpses of 'spiritual world' as well.
One of my dearest ambitions in 90s was to go to Moscow (I would not have been allowed to go alone, seeing as I was too young, and we lacked means to afford the proper trip) and participate in gatherings which took place in Neskuchny Garden every Thursday, which 'tolkienists' (the term 'tolkienutiy's' applied to themselves) termed Eglador. Several years latere there was much disquiet, as Eglador turned by and by to become what is now known (or was known in 2000-2001, I don't know if all activity hasn't diet out by now) as 'Poganishe' (Foul Place) - as majority of original members dropped out and instead of poetry-contest/musical/criticism/discussion gatherings Neskuchny showed tendency of having drunk youngsters impersonating orks and 'dark forces' on Thursdays.
Intensity of 'Tolkien Craze' during the 90s was such that there was actually a book written 'from another side's point of view' (the feat David Brin only toyed with the idea of) - somoene under the name of 'Nienna' published book named 'Dark Wings of the Wind' (if my memory is not failing me), which is supposed to be Silmarillion from Melkor's position.
Again, intensity was such that another chap, Nick Perumov, has written the whole series of so-called 'original' books, the first of which is a feeble essay in writing 'what happened next', as the hero is the hobbit and the events take place some 3 centuries after the War of the Rings. He made a real mess-up of it, though, as he tried to become more original.
There is quite a number of sites dedicated to Tolkien in Russia. One of them I know of (Arda-na-kulichkax) has a number of people who participated in original Tolkien movement, and grew up to become real good poets and writers and scholars (basing number of doctorates on Tolkien, even).
I suppose the release of the movies revived the interest, but I have no data to account for what happened over last several years. What I know for sure it there is not so much fuss in media about 'escapists' and 'satanists' of late.
As for Georgia, the Hobbit was just published some two or three years ago, the LoTR was not translated ever, and the movies where just movies. Not much of what should be called 'craze', rather number of people, mainly from what maybe called
intelligentsia, with good reading Russian infested with some of Russian craze, but nothing wide-spread. Tolkien was lucky personally for me, as my examiner in University entry examination was in love too, and I was lucky we've found it out, but that's another story.
PS I promised not to evaluate, but based on my memory of once made 'Map of the Barrowdowns', there were no members from Muslim countries. But I know, based on first hand experience, that there are Tolkien lovers in Azerbaijan. But again, as in Georgia, those are mainly people with good reading Russian catching the infection from Russian craze