That’s a hard one considering that H. W. Bush was a political lightweight who tried and failed to make it into the Senate and only really arrived as a major Republican Presidential candidate and then Vice President because people in the GOP were trying to unite around an anti-Reagan moderate (him being made VP was to bring the moderates behind Reagan) and only in the mid to late ‘80s did Bush Sr. even become a conservative.
There is also the issue of him being born in 1924 while his son, Dubya, was born in 1946.
With a PoD going as far back as a year after WW2, it’s hypothetically reasonable for Bush Sr. to reach the presidency anywhere between 1964 (age 40) to 1996 (age 72), no matter how unlikely the ends of that range are. Following those same rules, Bush Jr.’s first potential election would be 1988 (age 42) and his last would be, funny enough, 2016 (age 70, same as Trump and just barely older than Clinton).
That would mean that you get a pretty small window of realistic electability that can maybe be increased if Junior can somehow win in ‘84 at the age of 38 (it’d have to be a pretty spectacular election to get somebody younger than Teddy Roosevelt in the White House).
But if you were to do that, there’d be almost no way anybody would vote for the youngest President ever’s elderly father in ‘96 and very little chance in ‘92.
Now an easier way to do this would also bring some of its own complications. Lets say that with an earlier PoD, Bush Junior is somehow a boy wonder political figure in his 30s and wins a close election in ‘84 as the first Baby Boomer to be President and the youngest president by far (Reagan was born before WW1 and Junior was born after WW2 for some context). Whatever the PoD is, we’re handwaving the anti-nepotism moves made for the cabinet followed by RFK’s pick as JFK’s Attorney General and Junior picks his dad as his Secretary of State.
Following some sort of scandal (perhaps related to his history of struggling with alcohol or some sort of events take place that result in it becoming public knowledge that his administration sanctioned torture), Bush resigns and his VP is serving as a caretaker President until the next election when tragedy strikes at the 1986 State of the Union and the caretaker President, the House Speaker, and the President Pro Tempore all end up dead while Bush Sr. was off on a diplomatic mission.
Suddenly being thrust into the Presidency less than a year after his own son resigned it, Bush Sr. would remain an incredibly controversial figure and I couldn’t even imagine the conspiracy theories that would surround the Bush family.