To begin with, Tolkien writes in letter 142 that
I love music but have no aptitude for it [...] Slavonic languages are for me almost in the same category. I have had a go at many tongues in my time, but I am in no ordinary sense a 'linguist'; and the time I once spent on trying to learn Serbian and Russian have left me with no practical results, only a strong impression of the structure and word-aesthetic
which implies that large portions of his languages are probably not structured on Russian. I then further point to this essay. In summary, the
conclusion that Tolkien “consciously or unconsciously achieved a literary effect in terms of Slavic culture through the prism of the Germanic”, goes too far on too little evidence [...] Better would be to say that Tolkien did indeed incorporate a few Slavic elements into the multifaceted and multi-sourced structure and background of Middle-earth, but that he did so quite sparingly and, in almost every case, only at its furthest margins.
While the detailed analysis can be found in that article, it seems fairly clear that while a few words may have crept in here and there, these words were also Germanic cognates or were cut out of Tolkien's language, with only one or two exceptions.