Oh, it's absolutely the case that folklore emerges and changes rapidly - it's not likely much of our current folklore has direct origins that go back to Anglo-Saxon or Celtic times, but the recurrence of certain specific stories and ideas suggests a pretty widespread ancestral European belief in certain basic themes. One thing that emerges that the division between Germanic peoples and 'Celts' - which is based largely on artefacts and linguistic differences - is probably not very meaningful as far as cultural mythology is concerned: all the localised differences from and elaborations on the broader European themes are far more recent than the 'split' between these groups. "Celtic" in particular is not a very meaningful term in the ways it's popularly used today - at least in the British Isles, the term is used to encompass three or four loosely-related groups and tends to neglect mainland cultures that are closer to the Welsh and Cornish than those cultures are to Scots or Irish.
If you're interested in way folklore is transmitted and adopted, the Folklore Podcast has good coverage of several recent to very recent folkloric ideas including Slender Man (early 21st Century), chupacabra (1990s) and the bunyip (19th and 20th Centuries).
Celtic fairies and Welsh fairies aren't necessarily the same thing - it does seem that fairies are not traditionally a significant part of Welsh tradition as they are in England, judged by such evidence as landmarks named for them, or regional differences in belief that suggest a long local history of divergence and are all but absent in documented tales from Wales. This of course carries the caveat that this is what we have from documented interviews since folklorists started collecting stories, which is a pretty short period by the standards of folklore. Local fairies with older origins could certainly have been supplanted by English ones if there was no strong cultural attachment to them.
That does rather run against the point of creating a mythology for the English - their Christian-era beliefs are relatively well-documented. Snorri Sturluson was a Christian but what we know of Norse mythology, outside some references to practices in the sagas (themselves Christian-era retellings of oral traditions), is almost entirely drawn from his codification and retelling of stories that may have been treated very differently by the people who actually believed them or told them on a day-to-day basis.
Again, that presumes a continuity to the presentation of these entities that isn't really there in folklore, at least as far back as we can go. 'Sidhe' is just the Irish word for 'fairy' and is understood to include a class of folklore creatures as varied as the English word. The Victorian creation of small winged beings with a penchant for flowers has nothing to do with English folklore, but has led people into thinking that 'fairies' and 'elves' are different classes of being. All the cultural traits Tolkien gave his elves, and his presentation of them as a distinct stratified society, and that are synonymous with the modern notion are his own
Do you know of specific sources for this? I always imagined they had an older existence than Tolkien but haven't found anything concrete when I've looked (admittedly not in great detail). There are talking trees, but not perambulating ones that I'm aware of. Treefolk seem to be one of those things that feel as though they should be part of a real, older folklore, along with some other of Tolkien's inventions - I recall when I was younger I imagined that Old Man Willow and the trolls in the cave from The Hobbit were actual stories from folklore adapted by Tolkien, but they seem to have been his creations.
