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.
There is some confusion here between proper and improper nouns here. "metro" is widely used in Britain is a similar sense the USA to refer an urban railway or train service with frequent trains and stops. "subway" is likely to be understood in it's American sense (metro with a significant underground section) if there is sufficient context to exclude the British sense of a pedestrian tunnel.

A "subway station" could be either an metro station that is underground, or a station on a subway line.

"(the) London Underground", "the Underground", and "the Tube" are proper nouns and refer to an organisation (currently part of Transport for London, part of the London government) and the railways and train services it runs. "tube" may also refer to the small diameter trains and tunnels used on some London Underground lines.

"the London metro" is undefined, except for the Docklands Light Railway there is no set of railway lines with stops closer together, High Speed 1 is the only dedicated express line which not next to a metro line. See an [old geographical](https://www.whatdotheyknow.com/request/224813/response/560395/attach/3/London%20Connections%20Map.pdf) map or the [current plan](http://content.tfl.gov.uk/london-rail-and-tube-services-map.pdf). Compare with [Paris](https://www.ratp.fr/sites/default/files/plans-lignes/Plans-essentiels/Plan-Metro.1555581459.pdf), [Brussels](http://www.stib-mivb.be/irj/go/km/docs/WEBSITE_RES/Attachments/Network/Plan/Net_Reseau/Plan_Metro_Train_1902.pdf), [Hamburg](https://www.hvv.de/resource/blob/3204/d713455052fa2510b3f9cc5ee938c0ac/hvv-usar-plan-data.pdf), [New York](http://stewartmader.com/subwaynynj/) and [Washington DC](https://upload.wikimedia.org/wikipedia/commons/0/00/Washington_Metro_diagram_sb.svg).

Since London has no metro, can it have a subway? Possibly, but there are railway lines outside of London Underground with significant underground sections including the East London Line (since 2010), Elizabeth Line (under construction) and Waterloo & City Line (before 1995).

In summary, you can call many London Underground and some non-London Underground station "subway stations", but London Underground is not the "London Subway" and is definitely not the metro.

The reason the journeys in central London often involve the Underground, is that it and the Thameslink route are the only railways that cross central London.