To remount an old hobby-horse.
The Constitution is amended in 1820 so that each Congressional district chooses one Presidential Elector, with the others being chosen as the State Legislature decides. This gives a purely sectional party like the Republicans a far higher mountain to climb where electing a President is concerned. So a lot of Northern pols who became Republicans OTL, now decide that the Know-Nothings are a better bet and join that party instead. That doesn't guarantee a K-N win, but puts them in with a chance.
I am of course assuming here that the Mexican War and Kansas-Nebraska Act are not butterflied away. If the KNA is, then the Republican Party may never even get founded.