M.R.
Added: I'm sorry that not everyone's reading your post thoroughly. Folks, the OP did not say that her older child melted down until fourth grade -- it was mom, herself, as a child who had that difficulty. So no, the sibling didn't model this for the little girl; that is a misreading. And mom did not say that nanny has an easy time with dropoff either, though she doesn't detail what happens when the nanny drops off.....Telling kids to "shut up" as someone suggested is just plain mean, to a child or to an adult. -- OP, talk to the counselor ASAP and meanwhile let the nanny do all drop-offs (for consistency). Make your own departure from home for work very brisk and don't linger no matter what. I want to add too -- please do not ever bribe or reward in this situation.
Original:
I say this a lot on MP -- please e-mail the school counselor today. Don't hesitate. This is exactly the kind of thing the counselor is there to help you and your daughter handle. Yep, in preschool, the teachers and aides are great at distracting kids-- but in "real" school, the teachers have zero time to deal with meltdowns at drop-offs, frankly, and that's why it would help you and your daughter to involve the counselor immediately.
The counselor should be able to give you some ideas about how to make dropoff smoother and swifter (it has to be very brief) and the counselor also can touch base with the classroom teacher to ensure your child's tears through the day are not at the point of being disruptive to either her own learning or to other kids.
The teacher has said your child cries in class; how extensive is this? Does she cry for a short time after dropoff and then is actually OK the rest of the day, or does your daughter cry and mope on and off all day long? Is the teacher finding it hard to engage your daughter in ANY activity, or is your child OK with most activities? If the tears go off and on all day, and/or the teacher finds it difficult to get your daughter's attention for even the most fun lessons -- you definitely need to get some help via the counselor.
The counselor also should see your child one on one, weekly or maybe twice a week, here at the start of the year for a while. A trained counselor should have seen this before and should have engaging, fun ways to help your child gradually cope better. I'd see the counselor yourself, without your daughter there, the first time-- today, if you can arrange it -- then ask the counselor to spend a few sessions with your daughter working on her anxiety. The counselor is not a shrink, but IS very familiar with this exact issue in kids this age.
You do not mention whether your daughter went to kindergarten at all, or whether she went to K at this school or is new to the school. If this is her school where she went to K, I would really want to find out why she's so freshly upset -- she should be used to things by now. If it's a new school, that explains some of the upset, but not all of it.
The fact you have a new nanny could of course be part of it. I do get why you and your husband both tried dropoff in nanny's place, but that may be backfiring because your daughter doesn't know what to expect from day to day. All things to bring up with the counselor.