Well, Hiram Maxim apparently designed a semi-automatic pistol in the 1880's...
Say shortly after this, as the US is beginning to look towards replacing it's Colt revolvers. Now in OTL the single action revolvers were replaced with double-action revolvers. Perhaps the POD might be someone having a stroke of brilliance and recognizing the limitations of the revolver. Turning to Maxim they order a few of his semi-automatic pistols made which are later refined by American gunsmiths to incorporate the .45 round.
Perhaps they end up looking something like t
this...