EDDIE51, as I pointed out in the second paragraph of my very first answer above, mentee is in the OED.
The last print edition of that dictionary was produced in the late 1980s, so any word that has come into use since that date is generally to be found in the online edition. Although mentee had existed in the USA since the mid-60s, it clearly had not made sufficient impact on the editors over here to make it into the printed work.
TCL, re the 'or' ending...the verb nominate gave rise to the agent-word, nominator. We might, therefore, have expected the persons so named to be nominatees, but they aren't, they're nominees. The 'at' and 'or' elements just disappeared in the same way as in mentee.
The Mentor/Telemachus relationship was the reason I referred earlier to our word, mentor, as involving not just teaching but also affection. None of the usual synonyms for mentor cover that point. I suppose we could use prot�g�, but that comes from beyond our shores just as mentee does. If the Pond is a no-no, surely the Channel would have to be also!