The Christian mystics aren't mystics, but occultists; mostly the members of the Golden Dawn. Educated Brits working with Jewish manuscripts in the British Museum. They may have easily avoided any actual conversation with living Jews who are not liberal minded artists and painters.
Yes, they very easily can work without Jewish Mystics, since a) there aren't very many in England and b) they are not going to talk to middle-class British occultists. Jewish Mystics are dodging pogroms in Eastern Europe and are not going to share their maligned and persecuted religion with non-Jews; double that for female non-Jews.
Tarot, the deck and the game, was in existence for a couple of hundred years before French non-Jewish authors assigned the Hebrew letters to the major arcana. This was on the basis of the coincidence of the number 22.
Let me put it this way: of all the Tarotists here, including those interested in Kabbala, how many have entered an Orthodox synagogue and discussed Kabbalah with a Jewish mystic? And our time is much more liberal with respect to social contact between faith groups.