When Aziraphale's book store is first seen, there is a hat on the coat hanger where he hangs his coat. This hat belonged to one the novel authors Terry Pratchett, whose dying wish to fellow author Neil Gaiman was for him to write the series.
The fate of the surplus baby (son of the diplomat) is given by the hope that he'd be a happy child, yet instead would win prizes for his tropical fish. In the book, he became Greasy Johnson, a local village bully who was ignored by the TV adaptation.
Three of the nuns are named Sister Mary Loquacious, Sister Maria Verbose, and Catherine Prolix. The words loquacious, verbose, and prolix are all synonyms for "long-winded, talkative, or wordy".
When Crowley is approaching the ambassador's house to apply for the Nanny's position, (s)he is carrying an umbrella with a handle in the shape of a parrott's head, which is almost identical to another famous nanny's bumbershoot; Mary Poppins.
This episode (and to some extent the whole series) references extensively The Omen (1976), in which an American diplomat (Gregory Peck) is given an orphan child as replacement for his stillborn son at a hospital that subsequently burns down in a fire. The orphan, Damian, soon revealed to be the Antichrist, is rapidly seconded by a menacing rottweiler.