I'd figure things would progress much the same from the missed purchase to after WWII, unless the Russians managed to sell Alaska to someone else, like the Japanese or Canadians.
WW2 is where it gets interesting.
I could see a majority of Lend/Lease equipment going by rail to Novy Archanelsk in Alaska instead of by sea, increasing Russian presence in Russian America.
After 1947, with nuclear-armed Soviet bombers stationed right up on the edge of the western United States, most of the Cold War confrontation might not have happened due to the overwhelming advantage the Soviets would have. Berlin, Eastern Europe, Greece, Soviet intervention in Africa and other places would have been much more likely to have come out best for the Soviets.
Meanwhile in the USA, HUAC, McCarthyism, blacklists, and loyalty oaths, Civil Defense, bomb shelter salesmen, nuclear war drills in schools, all were a fact in OTL. With a real Soviet menace on the border... all of the most-probable paths I can come up with are very ugly.
To me, the Alaska Purchase timelines are fascinating because nothing much might change for 80 years, then the divergence would be abrupt.