Well, the “Catholic nuns” thing is pretty dubious, but veiling does pre-date Islam by quite a bit and was a feature in some earlier Christian thought, most famously that of Tertullian in the 3rd century A.D… His writing is rather illuminating to the later practice in Islam, actually:
*1] But we admonish you, too, women of the second (degree of) modesty, who have fallen into wedlock, not to outgrow so far the discipline of the veil, not even in a moment of an hour, as, because you cannot refuse it, to take some other means to nullify it, by going neither covered nor bare.  For some, with their turbans and woollen bands, do not veil their head, but bind it up; protected, indeed, in front, but, where the head properly lies, bare. Others are to a certain extent covered over the region of the brain with linen coifs of small dimensions----I suppose for fear of pressing the head----and not reaching quite to the ears. If they are so weak in their hearing as not to be able to hear through a covering, I pity them.  Let them know that the whole head constitutes “the woman.” Its limits and boundaries reach as far as the place where the robe begins. The region of the veil is co-extensive with the space covered by the hair when unbound; in order that the necks too may be encircled. For it is they which must be subjected, for the sake of which “power” ought to be "had on the head: "the veil is their yoke.  Arabia’s heathen females will be your judges, who cover not only the head, but the face also, so entirely, that they are content, with one eye free, to enjoy rather half the light than to prostitute the entire face. A female would rather see than be seen.  *
Emphasis added, from here: http://www.tertullian.org/anf/anf04/anf04-09.htm#P545_113997
So what we have is a cultural feature of pre-Islamic Bedouin society, pagan as well as Christian and Jewish, that got incorporated into the corpus of Islamic thought - though not directly, exactly. The Qur’an does not require veiling per se, but rather “modest dress”, with the covering of the hair being specifically mentioned ( a modesty feature common to Christianity at the time as well ).