Before we could return to the UK we needed to obtain a birth certificate and passport for Emily.
The hospital were very helpful in helping us navigate the birth registration process, which was completed with 72 hours of the birth at a local registrar's office. They also directed us to passport office at a nearby university to help us with the passport application process. Having been born in the United States, Emily is a US citizen by birth, and entitled to a US passport. However, even on an expedited basis a passport can take 3-4 weeks, and the process can be delayed by any errors in the paperwork.
We were fortunate to have the assistance of the office of senator Richard Burr, who liaised with the relevant authorities to expedite Emily's passport, so that it was processed in a little over two weeks, allowing us to fly home on our scheduled return date.
Before we flew, our UK lawyers provided us with a letter to explain Emily's immigration status and the basis for our expectation that Emily would in due course become a British Citizen. This letter, together with a copy of Emily's passport and birth certificate was sufficient to persuade the UK Border Force to allow Emily into the country on a six month visa.