Aside from the fact that it specifically takes place in the 1950s, there is surprisingly little about DC: The New Frontier that would need to be re-written for it to be an Earth-1 story.

Flash's temporary retirement is an issue, but it could be made a very temporary thing, in between issues. 

Lex is depicted as a businessman, but he is not critical to the plot and could be easily re-written as a mad scientist. I would have him as one of the scientists called up by Batman to help find a solution to the Centre's attack in the end.

Just as before. I will go though the characters and address where this would fit in their chronologies. The series spans a wide spread of publication years, so I am focusing only on the final battle with the Centre, which takes place only hours after Hal Jordan's first appearance in Showcase #22

For Aquaman, this is between Adventure Comics #264 and Adventure Comics #265

For the Blackhawks, this is between Blackhawk #140 and #141

The Challengers of the Unknown are between the events of Challengers of the Unknown #9 and #10. It's still relatively early in their careers

For the Martian Manhunter, this is between Detective Comics #271 and #272. The reference to his existence being a secret is completely correct. It was shortly after this period that his existence was first publicly exposed. However, in the Earth-1 canon, the untold origin of the Justice League (from Justice League of America #144) had already happened by this point, so the heroes involved in that issue should have already known about J'onn

The Flash appeared last in Flash #108, and next in Flash #109. Barry Allen had only just begun his career at this point, with only about a dozen adventures

Adam Strange is between the events of Mystery in Space #54 and #55. He also is relatively new to the business, having only visited Rann 8 times so far

For Green Lantern, as mentioned, the main action takes place within hours of getting his ring from Abin Sur, making this his real first mission

Wonder Woman is between #100 and #101 of her series. Her skirt would have to be retconned, since she had already transitioned to shorts by this point.

Batman, Lex Luthor, and Superman all appeared last in World's Finest Comics #104. Luthor was jailed at the end of that story, so if he is going to feature in the story he would need to be temporarily released, since he is back in jail in his next appearance in Action Comics #257. Superman appears next in Superman #132, and Batman in Batman #127. One major change I would make to plot in to eliminate the idea that Batman is distrusted by the authorities. At this time, Batman was very much a beloved public figure.

Green Arrow last appeared in World's Finest Comics #104, and would next appear in Adventure Comics #265

Now for the harder ones. Captain Nathaniel Adams, although he fits pretty well chronologically, would have to be written out of the plot because he is an Earth-4 character. That's not a problem. Ace Morgan and Hal Jordan can fly the mission into the Centre on their own

the Sea Devils don't work chronologically, since they did not form until July 1960 in  Showcase #27. If they were essential to the plot I might try to finegal the date, but they aren't, so it is easier to just write them out.

Ray Palmer and Doc Magnus, of course, are here long before their first appearances, but that's fine, since part of the story is that this is before they begin their careers. Ray supposedly spent years working on his shrink ray technology, so it makes perfect sense that he has a prototype here

The worst scheme from a re-write perspective is the scene where the magical heroes are considering getting involved against the Centre. We see Dr. Fate and the Spectre considering getting involved, but then they are confronted by the Phantom Stranger, Billy Batson, and Zatanna who tell them not to. Where to begin?!

First off, If this is to be set on Earth-1, neither Dr. ate nor the Spectre could be present. Furthermore, at this time on Earth-2, the Spectre was still trapped in the body of Jim Corrigan. Billy Batson is on Earth-S, AND trapped in suspendium at this point. And Zatanna is still only 16 years old.

The Phantom Strange is OK. He last appeared in Phantom Stranger #6, back in 1953. His next appearance on Earth-1 was not until Showcase #80 in 1969. As far as I know, there is no in-universe explanation for this gap. Zatanna can be replaced with Zatara. I would rewrite this scene to show Zatara watching Lois Lane's news report and considering intervening, only for the Phantom Stranger to tell him not to.

King Faraday last appeared in World's Finest Comics #64, back in 1953. He was not seen again until Showcase #100. Still, he did eventually return on Earth-1, so his death would have to be rewritten. I think that is an easy fix. I might even mention that he is badly wounded, which would explain his long retirement.