Nurse Appleyard has a sailing ticket for the "Rangitoto", a liner of the New Zealand Shipping Company which sailed from London to New Zealand via Panama. The company's ships did not serve Australia and so would hardly be a choice for somebody who was emigrating there.
Near the end Churchill gives the nurse a signed copy of one of his books. The book, "A History of the English Speaking Peoples, Volume II" was not published until 1956, approximately one year after Churchill retired as Prime Minister, and several years after the events depicted in this film.
Churchill had no nurse during this time.
Churchill rejects sitting on a stool to give the Margate party conference speech, saying "I'm not sitting in a bar singing White Christmas." The film and song actually came out in 1954, approximately one year after the events depicted in this film.
Nurse Appleyard wears a "Royal College Of Nursing" Badge. She should also be wearing a "General Nursing Council" badge [Silver and Blue enamel], which would be her "State Registered Nurse" qualification badge [SRN].