The poem that Lord Louis Mountbatten recites is Rudyard Kipling's "Mandalay" (1890), which is set in colonial Burma (now Myanmar). It is voiced by a Cockney soldier who nostalgically recalls a Burmese girl he met in the titular city. The text was set to music in the early 20th century and performed by popular artists, including Frank Sinatra and Bing Crosby.
The scene in which Queen Elizabeth II witnesses people testing the grass for minerals is an allusion to Kentucky bluegrass. As Bull Hancock explains in the scene, its high calcium content, ingested through grazing, is famous for producing gifted Thoroughbred racehorses. Claiborne Farm is in the heart of the Kentucky bluegrass region.
There was in fact an Arab oil embargo in 1967, a direct result of the Six-Day war in the Middle East. It lasted less than three months.
The race that Queen Elizabeth II attends is the 1966 Queen Alexandra Stakes, traditionally the last (and most prestigious) race on the final day of the Royal Ascot meeting. The race was won by Panic, Apprentice (owned by Her Majesty) finished second, and Valentine Day finished third. Apprentice would return to the race the next year, with the same result: second place, this time to Alciglide.
Early in the episode during a cabinet meeting, Prime Minister Harold Wilson (played by Jason Watkins) paraphrases U.S. President Lyndon B. Johnson when, referring to Louis Mountbatten (Charles Dance) and his meddlesome associates, Wilson says, "It's better that they're inside the tent pissing out, than outside the tent pissing in."