LOL RRM love the description, I'd hate her too .
My DD is 14 months and I found the first year tough [hopeful that it gets better emoticon]. Not only are you dealing with sleep deprivation on a scale you didn't know existed but everything you have worked towards in terms of your own personal happiness and comfortablness within your own skin is completely shot to pieces when you have a baby, Everything you were is gone and the feeling of 'being out of control' was, in my experience, the biggest, most brutal and life altering part of my struggle to cope certainly in the early days although on occasions I still get it today.
(IME) Not only do you look and feel different (and you don't like it) but you're totally knackered so can't be bothered to do anything about it. Sleep deprivation means you have no patience and can't be bothered with whether you'll upset someone so abandon the usual social niceties. There were times when I didn't have the energy to a) repeat myself b)use more words than were absolutely essential to get my point accross so was bloody rude especially to DH. In a nutshell you just shut down and try and stay alive.
If you're looking for some advice read on, if not feel free to ignore. I don't want to poke my nose in and I'm really sorry if it's way off the mark, it's hard to tell with what I 'know' of you.
I think, reading between the lines, there are a couple things you need sort out. Firstly, your DH. Is there a chance that your DH may be getting frustrated because he can see what the sleep deprivation is doing to you? My DH was always suggesting leaving DD to cry but ultimately it was about me, not coping and him clutching at not very imaginative straws. Could you express and have the weekends off or something? or do alternate nights? even if just for a week. If one of you is chronically sleep deprived and the other having lots of lovely sleep then there is a lot of resentment (well ime!) once DH took over some of the night times our relationship improved partly because I wasn't so tired, partly because he was taking more responsibility and partly because he could see where I was coming from. You may be reluctant to let go if you're the sort of person that likes to be in control, I know I was but honestly it will help.
Secondly, with the nanny. personally I think you need to sit down and have a chat with her when there are no others around and just be honest, tell her you feel crappy, knackered, jealous etc. If you tell her she may understand and even if she doesn't she might be able to sympathise. I'm a lay myself bare kind of person ()in the sense that if you tell someone what's going on then they are given some opportunity to redress the balance.
I think the three of you need to come to an arrangement about 'managing' her work. It shouldn't just be left to you. If your DH is a people manager then maybe you should trust him to manage the relationship and you decide her tasks together and if she doesn't perfom you decide how to deal with that together. She isn't just working for you and it's unfair of your DH to not take some responsibility for the realtionship.
Obviously you've got other issues with work and stuff but in the short term you need to do something now to get some sanity back in your life. I hope that is helpful and not patronising. Obviously it won't make you DS sleep any better but maybe if you were all more relaxed it would help him too!!