So WI George H.W. Bush or even Bob Dole had won in 1980 instead of Reagan, would the Cold War and the economy have gone any differently?
Probably no great changes in the cold war. Have read that the initial military build up under Reagan wasn't massively greater than Carter was committed to. After the invasion of Afghanistan detente was basically in a coma if not dead so there would be continued tension. Probably no SDI/Star Wars programme which would have possibly made it last a bit longer but the Soviet empire was already starting to stagger prominently. The big issue is does Gorbachev end up as Soviet leader or some hard liner? If the latter it could end with a bang rather than a whimper - which could be very bad for everybody.
If Bush [say] didn't feel pushed to adopt Reagan's hard line economic policies then you could see a more successful US economically. Definitely won't have something like Reagan's notorious chips speech. However the pressure for a right wing reaction to Keynesian economics and its more broad social policies was under way. It had already started in Britain with Thatcher so you could see a later President, almost certainly but not necessarily Republican, bringing a more elitist policy into play. However over 40 years so much could have happened in so many different ways.