I have to admit I don't know the politics and culture surrounding Nannies. Is it possible that the director doesn't either? Just because other parents with nannies know that it is taboo to "hire away" some one else's nanny doesn't mean that the general public does. I wouldn't see any problem at all of asking a neighbors nanny to also work for me if I knew she wasn't working during the hours I needed her. Is this possibly a case of a difference in culture rather than a deliberate under handed maneuver? If that is possible, than i think a FYI conversation with the director would be a good thing, to let him know that what he did was generally frowned upon in the nanny/client world. I wouldn't be rude about it, just informative. If I were the director and had inadvertently made such a huge faux pas I would want to know.
I think the perspective most of us are coming from is this-- would you expect your prospective new employer to ask your current employer if they can offer you a job? Especially if the prospective job theoretically wouldn't interfere with your current job and employer? I know in my husband's line of work he gets offers all the time and his current bosses are not contacted. Some times these are offers from a different group within the same company, sometimes they are from customers hoping to hire him.
I'm curious if you have told your nanny that you won't be enrolling your children at the gym because the director offered her a job?