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.
The author of the Saint novels, while not being American did live in the states for most of the period that he was producing the books, so it is possible that the particular usage you have picked up bled into the author's vocabulary during that time.

Leslie Charteris was born in Singapore to a Chinese father and English mother. He was educated in the north of England and briefly at Cambridge before moving to the US where he spent most of the rest of his life, so his familiarity with the niceties of the usage in regard to the London underground may have been limited by lack of exposure.

Of course, the line may be attributable to a script writer rather than Charteris, but even within the UK at that time the distinction between 'Underground' and 'subway' was, to an extent, peculiar to the *London* Underground. [*Glasgow's* underground railway](https://en.wikipedia.org/wiki/Glasgow_Subway) (the world's third oldest) has included 'Subway' in its name at its inception and currently.