Here’s a link to a website about serials. It used to be a discussion board 15+ years ago, but I don’t see the board any more.
The Serial Squadron Cinema Cliffhanger Archive
They’re on Fb too: The Serial Squadron Cliffhanger Library | Facebook
When was The Saint a movie serial?
1940s, I believe. There was another serial called The Falcon that was similar, but I don’t remember offhand if Sanders was in that one too. It may have been a spinoff of The Saint.
Whenever something was afoot, Sanders as “Simon Templar” would whistle to himself using the tune that later served as the outro for the TV series with Roger Moore.
The Saint and The Falcon weren’t serials in the sense of Commado Cody or Flash Gordon where you got a twenty minute chapter that ended in a cliffhanger and saw the next chapter the following week at the theater. They were both a series of full length B movies put out over several years - like the Blondie and Dagwood or Ma and Pa Kettle movies.
The current serial being shown on TCM on Saturday mornings is Holt of the Secret Service.
Those were not movie serials as described in the OP.
Definitely not the same tune: Roger Moore Version Bing Videos
The whistle George Sanders does at the beginning of this clip How George Sanders Brought Simon Templar as The Saint to Life on the Big Screen
But the George Sanders theme did match up with the beginning of the theme of the Ian Ogilvy Return Of The Saint series Bing Videos
That’s the tune to which I was referring. It can be heard in the YouTube clip “The Saint Theme Late 1960s ‘In Stereo’,” whose link this site is preventing me from posting. 
There were different versions of both the intro and the outro between 1962 and '69. I can’t tell which is in the clip because most of it is taken up by pictures of Roger Moore.
But you do know the difference between a movie series and a movie serial now, right?
I knew it before. I just remembered Sanders’ movies, which I saw several years ago, as belonging to the latter category.
None of the Sander’s Saint movies were serialized as far as I know.
A 1940s Zorro serial was included in the KSTP marathons. In one episode, a hacienda or other large building was blown to smithereens as part of the cliffhanger. It magically reappeared toward the end of the serial as Zorro was rallying his compadres, since no one expected the audience to remember it had been destroyed a few weeks before.
Apparently not.
I was too young to see any in the theaters, but one of the local TV stations had a block called “Matinee at the Bijou” on Saturday afternoons, when they would play a couple of old movies, and a chapter of a serial. I remember The Crimson Ghost and The New Adventures of Tarzan.
They released a few serials on VHS, but it was DVDs that really enabled me to indulge.
I have Buck Rogers, and all three Flash Gordon serials. I was struck by the actor who plays MIng the Merciless. In the first two, he just chews the scenery. But in the third serial, his acting skills improve significantly.
The Adventures of Captain Marvel shocked me twice. In one scene, a group of gangsters attack Captain Marvel on the roof of a skyscraper. Captain Marvel picks up one gangster, and casually tosses him over the edge of the roof. You couldn’t do that in a kids’ movie today! In a later chapter, the heroes are attacked by a horde of bandits. Billy Batson mans a machine gun and mows them down in droves. You couldn’t do that in a kids’ movie today!
The Phantom Empire stars Gene Autry, as a singing cowboy, in a science fiction story. (No, I am not making this up.)
I just found Canadian Mounties vs Atomic Invaders on YouTube. Sadly, it’s an espionage flick, not a sci-fi flick. But I still intend to watch it.