Are there any reasons they should be considered The Regime earth? I don't think they should be related to the The Regime universe. In game we are told that they are from the multiverse. I propose to refer them to the universe "Injustice: Gods Among Us".
1) Jay Garrick. In game during the intro for mirror match, he says that he is stuck on this earth. We conclude that he is not from The Regime earth. However, we don’t know what year he came to The Regime earth, so he could be a member of the Justice Society and we don’t need to have a separate page for Jay Garrick who was mentioned in comics.
2) Jesse Quick. Regime's Jesse Quick dies in Injustice: Gods Among Us Vol 1 7.
3) Wallace West. In mobile game there is already a red-haired Wally West, so if move will need to add R. to the page name.