I personally wouldn't risk it - as you are aware, it's illegal for a US citizen to enter the US on a foreign passport, and you could have trouble with US immigration if you try to enter without her US passport... especially if other members of the family have US passports and she doesn't, because they're more likely to question why she doesn't have the same nationality as her sibling.
You have a couple of months before you travel, and looks like it's currently taking about 5 weeks for the Edinburgh consulate to process the report of birth abroad and first passport (you have to submit the documents 7 days before the appointment), or if you can get to the London Embassy, it's taking 4 weeks (and you just take the documents with you to the appointment).