"women must "choose" between their professions and their children."
As a professional myself, I know EXACTLY how you feel. I totally understand where you are coming from. I am a first time mother too, and full time working mom, and I feel a lot of guilt at times for making the choice to work. I, too, have had my son push me away so that he can cuddle with with nana, who, at times, due to my work schedule, has taken care of him for weeks at a time.
Trust me, your daughter loves you more than she can ever express, because you are her mother, and the bond that you and she made when you were still "one" (in your womb) and when you breastfeed with NEVER go away. Period. Your daughter, however, also has love to give to others, and that is why she is so affectionate with the nanny. It really does not mean that she loves you less, it simply means she loves nanny, and perhaps she loves her more THAT day because nanny prepared a special meal, or nanny let her wear her favorite dress, or play outside for a little longer....whatever. Children are finicky. Although I am not quite there yet, one piece of advice that I heard a while back...children, because they have no "internal filter," and are so pure at heart, are sometimes so honest and truthful, sometimes it can hurt your feelings. It's important to not internalize that, and to realize that she is child who does not really comprehend the full scope of what she is saying. She does not mean to hurt you. If she really understood that she was hurting you, she would stop.
Ultimately though, this letter sounds a little like a mixture of jealousy and possibly guilt over the choice of being a working mother (but I don't want to sound harsh or anything, I am jsut guessing and I clearly don't know you well enough, so I apologize if offense is being taken here). If the former is true, remember that you are your daughter's mother, and that is the most coveted position of all! You will influence your daughter's life in more ways than a nanny ever, ever will. It's that magical bond...it's really representative of a love that cannot be defined in words, and a love that is neverending. NO nanny can give that to a child. If the latter (guilt over working) is true - then, remember that you chose your profession because you respected your own talents enough to cultivate them into something meaningful, something that impact the lives of others. This is a wonderful example to show a little girl! If working did not feel right, you would not be doing it. However, it's satisfying and rewarding to you, and that is important. Who ever made up the rule that you ought to sacrifice your pursuit of happiness for the sake of your child??