First off, I decided to merge the continuity of Earth-1 and Earth-2 for this, since if you leave out all the characters not native to Earth-1 there are very few left at all. I only eliminated the characters from Charleton and Quality comics

The miniseries is set between the formation of the JLA told in Justice League of America #9 (which happens around October, 1959 in publication dates) and Brave and the Bold #30, so I am focusing on the characters that would be available at that time.

The largest retcon is that Black Canary needs to be switched back for Wonder Woman, which would involve heavily changing most of the fight scenes among other things. The other one is the conceit that Superman and Batman did not join the JLA after defeating the Appellaxians, and did not become members until after the defeat of Amazo in Brave and the Bold #30. I actually have no real problem with this, even though it involves changing the Silver Age continuity a bit, since Superman and Batman played little role in the JLA's Brave and the Bold adventures.

The following heroes would not be available in the time frame in question; Animal Man, Cave Carson, the Doom Patrol and the Brotherhood of Evil (sadly), the Metal Men, Metamorpho, and the Atom (Ray Palmer) all had not yet begun their careers. Wing was dead. The Crimson Avenger, the Vigilante and the Shining Knight were all trapped in the past. The Spectre was still trapped in Jim Corrigan's body. I'm fairly sure Robotman 1 was still trapped at the bottom of the mineshaft, and that Johnny Thunder was out of touch with his Thunderbolt.

The other golden age heroes in the story were all either retired, or semi-retired at the time, but still available to participate in stopping the invasion; Amazing Man, Black Canary, Dr. Fate (presuming he had already recovered the Helmet of Fate by this point), Dr. Mid-Nite, Dr. Occult, Green Lantern (Alan Scott), Hawkgirl, Hawkman, Hourman, Johnny Quick, Liberty Belle, Mr. Terrific, Plastic Man, Sargon the Sorcerer, Starman, Tarantula, the Atom (Al Pratt), the Flash (Jay Garrick), the Sandman, Wildcat, Zatara the Magician

Technically, all the members of the Global Guardians (Bushmaster, Dr. Mist, Godiva, Jack O'Lantern, Owlwoman, and Rising Sun) did not appear until the 1970s. However, since we have no specific information on when they began their careers (and some anecdotal evidence that Dr. Mist founded the Dome in the late 1950s), I have no issue with their being active at this time.

The Sea Devils debuted right around the time of the full-scale Appellaxian invasion, so I have no issue fudging the date a bit so that this is a month or so after their first appearance.

Between the formation of the League and the battle with Amazo, Aquaman had 9 appearances, from Adventure Comics #265 to Adventure Comics #273

Batman had 33 appearances, from Batman #127, to Batman #132

The Blackhawks had 30 appearances, from Blackhawk #141 to Blackhawk #150. The miniseries shows their temporary shift to becoming more traditional superheroes, but this is anachronistic and would have to be written out. It happened much later.

The Challengers of the Unknown had 9 appearances, from Challengers of the Unknown #10 to Challengers of the Unknown #14

Green Lantern (Hal Jordan), interestingly, had only 2 appearances during this period, both from Showcase #24. He was not very active during the early years of forming the Justice League. In fact, by the time of Justice League of America #1, Hal had only had about half-a-dozen cases, outside of his adventures with the team. Of all the heroes, he was the biggest rookie, even if he was the most natural leader.

The Martian Manhunters existence had just been revealed to the world, shortly before the Appellaxian invasion. He had seven cases during this period, from Detective Comics #274 to Detective Comics #280

Green Arrow and Speedy had 11 adventures during this period, from Adventure Comics #265 to World's Finest Comics #110

Superman had 66 adventures, from Superman #132 to Action Comics #265. He was certainly the busiest superhero during this time. It makes a lot of sense that he didn't have time for the League. Batman is the only one who comes close to his score, and he had half as many adventures during the same period.

Flash (Barry Allen) had 5 adventures during this period, from Flash #110 to Flash #113

Wonder Woman had 6 adventures during this period, from Wonder Woman #109 to Wonder Woman #114

As for the villains; Clayface and Solomon Grundy don't work, since Matt Hagen had not gotten his powers yet and Grundy was trapped on the moon. Eclipso doesn't work either, unless this is an earlier version preceding Bruce Gordon. The battle in the hotel would need to be changed to a totally different lineup of villains.

The League is also shown battling Killer Shark, Gorilla Grodd, the Invisible Destroyer, and the Icicle. All of these are possible. Killer Shark had been active for years (this would be between his appearances in Blackhawk #139 and Blackhawk #155). Grodd was recently imprisoned in Gorilla City by Flash in Flash #108, and would next appear in Flash #115, but he could have escaped and been recaptured in-between those appearances. The Invisible Destroyer first appeared just before the League formed, and the Icicle had been semi-retired for years.

The last villain to play a major role in the plot is Vandal Savage. Of course, he was active during this time (when is he not?)

Interestingly, a number of heroes also appeared during this time that were NOT included in the miniseries, which I think was a mistake. Aqualad and Kid Flash began their careers at this time, and would have to be referenced, and probably depicted in the series. The Elongated Man also debuted during this time, and I would have him among the heroes fighting the invasion.