I thought asking the opposite question might be interesting.

What if the likes of the FF, Spider-Man, Hulk and so many others had debuted in, say, 1938 or 1939? How different would their origins have been?

I'm not an expert on fiction from the 1930s, but, during that era, and well before, there certainly seemed to be scientists, good and bad, injecting themselves, messing around with compounds, experimenting on themselves and others. Perhaps a Reed Richards, circa 1939, might have injected himself with something that gave him powers (we'd have to assume the rest of the FF were willing volunteers, too).

Ditto Bruce Banner. No gamma radiation, just experimentation in a lab.

Would Tony Stark have been a WWII soldier who got to develop his armour on the battlefield?

Spider-Man is an interesting one. I can't think of an origin for him in that period - let's assume radiation wasn't heavily featured - so perhaps someone else has an idea.

Gods like Thor have pretty timeless origins.

Would Doctor Doom have been on the side of the Axis or Allies? Or would Latveria have remained neutral? 

Would mutants have been seen as cannon fodder and been the first to be conscripted?

Looking forward to everyone's imagination running wild!

The creators of comics back in the early days had different mindset regarding superheroes. They tended to be created by "science", magic or by being a perfect physical specimen.

Some idea of how the 60's Marvel would have appeared in the 1930s and 1940s might be found in the proto-characters the day.

Spider Queen first appeared EAGLE #2 Sept 1941

Thor first appeared AIR FIGHTERS #2 1942

IRON ACE first appeared WEIRD COMICS #1 1940

As mentioned, the Hulk might be closer to his Jekyll and Hyde antecedent and be made on a lab rather than a bomb range. 

As Spider-Man, if Bob Frank can get powers from a mongoose, then Peter Parker can still get powers from a spider bite. 

Quality had a strip that featured a robot called Bozo the Iron Man. The title character, Hugh Hazzard could climb inside him to fight crime. 

1. IRON MAN--Definitely Howard Hughes type who creates a flying suit (with some extra gadgets).

2. INCREDIBLE HULK--"Dr. Banner and Mr. Hulk" with (as Stan Lee originally envisioned) some of Frankenstein's Monster thrown in.  (Less jumping miles and fighting giant robots and monsters, more shadow and horror.)

3.SPIDER-MAN--Probably a mix of the Tarantula crawling on walls and the (bodysuit) Sandman with his cable gun.  He could be bitten by maybe a "magic" spider and actually gain powers, but the name leans more to a pulp fiction-type normal man crimebuster, albeit with gadgets.

4. CAPTAIN AMERICA--I know we're talking about 60's Marvel, but the idea of a 30's Cap is sort of fascinating!  I envision a Chaykin-ish Scorpion/Dominic fortune type outfit, but with aviator mask and shield.

5. DR. STRANGE--The Mysterious Traveler goes back to the 40's but push that back a bit and I'm picturing a Dr. Strange in hat and trenchcoat who shows up where needed, helping people (as in the first Nightmare story) who are tormented by supernatural forces.

6. THOR--More of a Shazam/Captain Marvel type transformation than we've seen.  I would even keep Dr. Don Blake with the cane.

7. DAREDEVIL--Well, the original Daredevil fits the bill pretty nicely, doesn't he?

The teams are tougher for me.  I think they'd all be different variations on Prof. Challenger and friends.

Doctor Reed Richard, the world's smartest man, called "Doc Fantastic" by the press, uses his fantastic intellect to fight crime from his skyscraper headquarters in New York City

Ben Grimm, Reed's best friend, ace pilot and barroom brawler. 

Johnny Storm, thrill seeker, engineer and practical joker. 

Sue Storm MD, Reed's girl friend, Johnny's sister. Expert in medicine and biology. 

From  hidden civilizations high in the Himalayas to secret caverns in depths of the Earth you can always count on the Fantastic Four. 

Thinking on Hitler: "...you arrogant swine... I shall do your bidding for now, because it suits my own experimental purposes. But no man, not even the leader of Nazi Germany, shall long control the destiny of Victor von Doom!"

Later, to Hitler: "Nazi dog! Just be glad I don't kill you myself!" "You are the one man on Earth who is more evil than I could ever hope to be!" And then pontification about his gypsy origins to Hitler (trapped under rubble.)

Notwithstanding that Doom would never think of himself as evil, I guess this indicates how he would feel about the Nazis pretty well.