Question: Here in America, I was taught in the mid-60s by disc jockeys playing the Petula Clark song that in the UK "subway" means a pedestrian tunnel beneath a street, not an urban rail transit system. But on today's rerun of "The Saint", an episode set in London, a character with a British accent says to Mr. Templar, "She committed suicide. She stepped off a subway platform right in front of a train." This episode is in color, meaning it was made around 1968-69. 

So can subway be used for the Tube as well?

Answer: 
Your understanding of the different uses of "subway' are correct. In the UK it means a passage (usually walkway) beneath something, often a street.

However with internationally marketed entertainment a different dynamic often comes into play. Whereas British audiences would mostly have understood the meaning of Americanisms, even in 1969, it was generally assumed that US audiences would not have understood the meaning of Britishisms, even if they were used in a strictly British context. Such shows often take the decision to use the American terminology even when it is illogical to do so.

For an extreme example consider the movie [Sliding Doors](https://www.imdb.com/title/tt0120148/goofs/?tab=gf&ref_=tt_trv_gf), which constantly uses American references ("Jeopardy", "Class One drugs") even though it is entirely set in Britain and virtually all the characters are British.
I cannot account for *The Saint*, but as a native of England I would find it very strange to hear another of my countryfolk refer the London Underground system as the *subway*. It would almost always be referred to as *the Underground* or *the Tube*.

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> Take the *underground* for two stops, but be quick as *the tube* station closes early on weekends.
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Subway in the UK tends to refer, as you say, to a path underground typically beneath a busy road system. Also referred to as a pedestrian *underpass*, with footbridges over busy roads often called a pedestrian *overpass* as an antonym.

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> If you don't want to cross through the traffic there is a *subway* you can use, or there's the overpass if you don't mind heights.
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