This is actually episode 7 of the season, rather than episode 13.
During the period this series is set, Heathrow was known as London Airport. It was renamed in 1966.
The French word docteur is consistently used as the translation for the English "[medical] doctor", but docteur is used only as the title or form of address. The proper translation for medical doctor should be médecin.
Using 1963 technology, the crew would not have been able to accurately line up with the runway with no directional beacon or any nav aids in the dark?
707's of that day did not have an onboard APU (auxillary power unit) to power interior lights and restart engines. (Yet it was clear during the episode that all engines had been shut down after landing and the interior lights were still on.)
The episode kept shifting from CGI views of turbojet engines (exit tubes) to under wing shots of turbofan JT3D engines at the airport.
"Break it to me gently" by Brenda Lee (at the end)
Goof: The policemen in Rio were driving an Italian Lancia police car. This was not the regular issue police car in the 60's - or ever! Maybe in Italy...
The Corcovado mountain with its Christ is not located at the end of the beach in Rio.
Colette sings the first verse of the German Anthem (from "Das Lied der Deutschen") at the U.S. Mission which is closely related to the time of the Nazi Regime (The first line of the song says "Germany, Germany above all").
Nowadays they only use the third verse as the official German Anthem (roughly translated Unity and Justice and Freedom).
Continuity: Laura has dyed her hair since the last episode. While this isn't a problem, it makes the flashback scenes to her wedding day look odd when she is suddenly blonder than she was before.