Bush likely will govern as a true fiscal conservative, rather than this Reganomics nonsense. Not that that will be significantly better for the economy as a whole (it will likely make it worse, with cuts and tax increases going into effect before the 1982 midterm elections), and the most likely result, I think, is a 2010-esque Democratic wave in the House, with the Democrats nabbing 60 some odd seats and a gain of 5 or so seats in the Senate, rendering the chamber back unto Democratic control.
The result is going to be a thoroughly emasculated President Bush. Like Eisenhower after losing the 1954 midterm elections, he'll likely tack to the center and try to focus again on reducing the deficit in a more liberal fashion. Unemployment, of course, should still be high, and the 1984 Presidential Election will probably be a lot closer than it was IOTL. Regardless, Bush's foreign policy credentials and image as a compromising centrist will probably help him out here, and he'll likely win re-election by a much smaller margin than did Reagan in 1980 over Walter Mondale.
If Iran-Contra goes ahead, don't expect Bush to go unscathed. He doesn't have the 'Teflon' qualities of Reagan, and he likely will face some sort of censure from the Democratic Congress prior to the 1986 midterms, which will look like 1958. The Democrats have themselves solid control of Congress going into the 1988 Presidential Election, which will feature Republican Vice President Bob Dole against Senator Ted Kennedy and Senator Dale Bumpers.
With the economy still not having fully revived at that point, Kennedy wins a close victory over Dole. The GOP makes gains in Congress, but in the end, the country votes for Kennedy and rejects the fiscal restraint of the GOP in favor of heedy fiscal stimulus. . .
Presidents of the United States
1981: Ronald Reagan (Republican) [1]
1981-1989: George Bush (Republican)
1989-1997: Edward Kennedy (Democrat) [2]
1997-2005: Robert Dole (Republican)
2005-2009: Matt Fong (Republican) [3]
2009-2017: Margaret Anderson Kelliher [4]
[1] Assassinated.
[2] First brother of a former President elected President.
[3] First Asian-American (Chinese-American) elected President.
[4] First woman elected President.
Vice Presidents of the United States
1981: George Bush (Republican) [1]
1981-1989: Bob Dole (Republican)
1989-1997: Dale Bumpers (Democrat)
1997-2005: Jack Kemp (Republican)
2005-2009: Mike Rounds (Republican)
2009-2017: Joe Manchin (Democrat)
[1] Became President upon the death of Ronald Reagan.