I would wager that the animals in the books are less those of Northern European mythology, but simply those of Northern Europe, which is the approximate location of the north-west of Middle-Earth, where the main stories take place. Flora and fauna would be influenced less by deliberate decisions of the author, but the climatic, geographic, and historic circumstances.
Some animals are different in appearance and temper because they were affected by Morgoth and turned into "monsters of ivory and horn" (passim throughout the later QS writings such as HomeX). These include, of course, both beasts from the north-west of Middle-Earth and Europe, and from everywhere else.
Animals with reasoning and speech are a different matter. Tolkien admits that "these have been rather lightly adopted from less 'serious' mythologies, but play a part which cannot now be excised" (MT).
Tolkien explains this further in Myths Transformed:
"Living things in Aman. As the Valar would robe themselves like the Children, many of the Maiar robed themselves like other lesser living things, as trees, flowers, beasts. (Huan.)"
However, this is at least partially contradicted by the later statement "In summary: I think it must be assumed that [orcs] 'talking' is not necessarily the sign of the possession of a 'rational soul' or fëa...talking was largely echoic (cf. parrots), in The Lord of the Rings Sauron is said to have devised a language for them.... The same sort of thing may be said of Húan and the Eagles: they were taught language by the Valar, and raised to a higher level - but they still had no fëar." (ibid.)
Some discussion on the matter can be found in the
Ealar and Incarnation thread.