"... I don't expect sweeping gratitude from my children's friends but I do expect basic manners and respect."
This. So many thousands of times, this!
I think that as your friend informed you that her precious son "doesn't like your nanny", it's pretty clear where he gets his entitled behaviour from. What was she expecting you to do? Sack your nanny because a child whom she wasn't employed to care for, doesn't like her?! And you were right in not passing that complaint onto your nanny - I wouldn't have either. Because basic manners dictate that we're better off not knowing such childish complaints, frankly.
One of my son's friends is actually banned from my house, because when he was 7, during a playdate, he behaved like a little dictator. He didn't like where our dining table was, for instance, because he couldn't see the TV (which was off, incidentally) whilst he ate, he wanted ice-cream not fruit for afters, thought it a good idea to kick his feet against my table leg whilst they ate, interrupted a conversation between myself and my (then 16 year old) daughter about her homework, lectured my daughter about how she'd chosen all the wrong 'A'-level subjects, she should do X, Y, Z instead, threw all my son's toys around in his bedroom (which took over an hour to clear up) because I wouldn't let them veg out on the sofa and watch crappy after-school television, and finally, for good measure, pulled the dog's ears so hard that he yelped.
When I marched him home (literally over the road from us), after the ear pulling (he was very lucky our old dog was too surprised, I think, to bite him), and told his mother what he'd done/how he'd behaved... her excuse was literally "oh, well, [Bob] does like to have his own way in things, you know...". That was great, I said, in his own house - but not in mine. In my home, I expect basic manners and common courtesy - and what I serve after a main meal is dependent upon what I decide, not what my then-7 year old son demanded. Less so, her precious little "Bob" (obviously not his real name).
The boys are now 16 and still friends. "Bob" was offended that I refused to allow him to step foot over my threshold for a while, I think (even now, he knows to stand in the porch whilst my son gets his shoes on, if they're going to the park with other friends), but tough. I was offended by his rudeness and his mother's attitude of allowing him to be thus (I would have been mortified had it been the other way round, and my son would have been told off, not cuddled by a doting Mama!).
Think of how your nanny must feel and have felt regarding this boy's behaviour. I'm willing to bet there's been more rudeness/bad behaviour that you don't actually know about. She may be unwilling to tell you, given that she must know the other mother is your friend, too.
If the boys are genuinely friends, then no longer having this other boy inside your home on a Wednesday isn't going to stop them from so being, is it?