[Scott]

Lalaith,
While you were right to say that Quenya was not based upon Brythonic languages (it was based upon Finnish, Greek, and mainly Latin) Sindarin however was based upon Brythonic languages immensely.

To quote Ardalambion,
"A Welsh- or Celtic-sounding language was present in Tolkien's mythos from the beginning. This language was originally called Gnomish or I·Lam na·Ngoldathon, "the tongue of the Gnomes (Noldor)"
and,
"In 1954, in Letters:176, Tolkien stated that "the living language of the Western Elves (Sindarin or Grey-elven) is the one usually met [in LotR], especially in names. This is derived from an origin common to it and Quenya, but the changes have been deliberately devised to give it a linguistic character very like (though not identical with) British-Welsh: because that character is one I find, in some linguistic moods, very attractive; and because it seems to fit the rather 'Celtic' type of legends and stories told of its speakers". Later, he found that "this element in the tale has given perhaps more pleasure to more readers than anything else in it" (MC:197)."
Since the language of the Irish and English oriented celts was much the same and stemmed from one another, I think that it isn't much of a stretch at all to say that some of Tolkien's languages were influenced by Irish languages, at least Sindarin was (and of course Ilkorin) was.