I have recently developed white eyelashes on half of one eye but nowhere else and I have no white patches on my skin - what is this likely to be?

Eyelash whitening. Eyelash whitening can occur for no reason at all but it can related to autoimmune disease. See an eye doctor for further evaluation.
White eyelashes. Known as vitiligo or poliosis, eyelash depigmentation (whitening) can be a result of some hereditary condition. My suggestion: see an ophthalmologist (not an optometrist or optician!). I do not think it's serious, but the eye doctor might be able to determine why and, perhaps, offer a way to reverse it back to normal.