If this were to happen the GOP would hold the Presidency longer than FDR and Truman did. I think it's unlikely and the Dems would likely take the White House back in '68 or at the latest '72.
I'd speculate that if a less-paranoid Nixon wins in '64 and continues with Ike The Sequel Plus Dick's Neat Ideas he is well positioned for re-election. Now imagine if Nixon gets it in his head to push universal health care and LBJ drools at the prospect, that could also make him decently strong.
Without Vietnam, all kinds of things that had been quashed have room to thrive. American SST airliners. Could one imagine high speed rail?
MLK alive will ensure that progressive energy in the national dialogue does not easily fade into the background.
After a while, Teddy Roosevelt might find it quite familiar.