As far as I can see the flowers used by Peter Jackson in his adaption of the LOTR books were the flowers of puawhanga (Clematis paniculata) (or possibly some other white-flowered Clematis) - irrespective a native vine and so not Tolkien's mythological plant. There is no such plant fitting his imagination present naturally in N.Z. - and I am a rabid Tolkien fan I assure you. Tolkiens description seems better to have been modeled on some Northern Hemisphere bulbous plant that he may have liked or seen or had fond memories off.
Clematis paniculata is often commercially available, it is easily grown, preferring a moist soil - this should not dry out but should not be waterlogged either, and planted in a place where it can grow up into the light. It flowers best with its head in the sun and its feet in the shade.