Its interesting how history often repeats itself. The 1952 Republican Convention was one of the nastiest in recorded history. I suppose you know that Robert Taft may have had the most delegates initially. However there was a lot of accusations of delegate tampering and putting his own guys in there over others, and in time the Convention censored him and removed many of his delegates from the South, as a result Eisenhower wins the nomination.
Nixon wasn't involved in those allegations so that's going to play out as it did. Nixon was chosen for his political experience and his strong anti communist views.
Given the allegations there's no way Taft is the VP in 1952 election. My guess Sen. William F. Knowland, he has many of the same characteristics Nixon did at the time. Especially about being against China and communism. He has the experience in politics that Eisenhower lacks and a working knowledge of the Congress. He however doesn't really have the fire of Nixon and would never have made something like the Checkers tape. He was reportedly Taft's number one pick for the VP and Eisenhower's second choice as well.
Knowland being in the VP in 1952 could have serious consequences, he was very much against the People's Republic of China getting the UN Seat that was once held by the Republic of China. What you could see is the potential that the Republic of China gets the seat and the People's Republic of China is expelled instead of OTL (Meaning Taiwan has the permanent seat and China has no standing in the UN).
I still believe Eisenhower/Knowland wins the election. It will not be by the landslide Eisenhower/Nixon got in the OTL, but it will still be more than sufficient for them to win election. I also believe they'd win reelection in 1956 making inroads to the south and winning by more than they did in 1952.
Without Knowland in the Senate in 1957 it brings to doubt whether the Civil Act of 1957 could even pass. Bill hated the bill and wept because it was so weak, but without his influence I'm not seven sure that bill would pass the Senate. VP has a lot less influence on his fellow Senators than being one of them.
In 1960 Knowland if all else is equal still loses to John F. Kennedy. What's going to kill Knowland is the Unions, he espouses Right To Work as a national right probably. That'll swing the election big time. Combine this with the failure of the Eisenhower/Knowland administration to pass a Civil Rights Act and well its just gonna tailspin.
In 1964 he'd support Barry Goldwater, Johnson much like in the OTL will win by a landslide due to JFK passing a Civil Rights Act and the mood of the times.
In 1968 his marriage is starting to suffer some strain so he probably doesn't run. He probably would not campaign for them but he'd probably support either Ronald Reagan or George Romney. Since there is no Nixon, I believe Ronald Reagan would be the Republican Nominee in 1968. I believe he would beat Hubert Humphrey and George Wallace much like Nixon did.
Bill unfortunately would commit suicide in 1972 with massive debts hanging over his head and his second wife leaving him finally.
Ronald Wilson Reagan, president in 1968 is going to change a lot. First off my guess is George Romney is Vice President in this scenario. There's not going to be a Watergate scandal.
Reagan's not the type to half do a war, Vietnam is probably going to end much differently than it did. Does it lead to a WWIII? Possibly, remember the strong feelings about denying People's Republic of China the UN Security Seat back in the Eisenhower/Knowland administration? They've not forgot......at the same time they also probably want to avoid all out war with the United States. What I see is where PRC gets the security council seat that RC once had and that hostilities end with far less damage to Vietnam overall than the OTL. As a result of the Reagan influence relations will improve over time I believe and it is possible just like Nixon did that Reagan will open up China to trade, but under far better circumstances.
Another thing the US doesn't go off the gold standard. Yep that's right the money is still backed by gold. (That's mostly because Nixon isn't president and Connally isn't treasury secretary as a result. Note that George H. W. Bush is still appointed in my TL here as Envoy to China).
Reagan/Romney most likely win reelection in 1972.
1976 Romney obviously is top seed, but whose his main opponents. I don't think any of the remaining possibilities are strong enough to beat his nomination. There is something odd though, I believe Romney would choose a moderate Vice President, as a way of promoting party unity. The most visible moderate, Gerald R. Ford, Jr. Odd isn't it.
Democrat ticket, there was no Watergate, therefore no rule change in how delegates are assigned, an unknown like Jimmy Carter will never become president. I believe Ted Kennedy would refuse to run in 76 for much the same reasons he did in the OTL (He will be looking to 1980 though). This is just a guess but with Ted sitting it out and no rule change, Robert Byrd of WV I believe would be the most likely candidate.
Romney/Ford win the 76 election. With not moving off the gold standard we do not have the rapid deflation then stagnation that the Carter administration knew. Iranian Revolution does occur, but they do not hold the Americans that long mainly because Romney/Ford is much more overt about their response.
1980 Ted Kennedy is ready to run, to reclaim the throne and to right many wrongs. He runs on a platform of health reform, civic reforms. etc. Without his war with Carter the previous years he has a more cohesive campaign message and a better ground game. He is able to secure the nomination. Unfortunately his past catches up with him, and while his message strikes many hearts chants of "Where's Mary Jo?" ultimately doom his candidacy. Romney/Ford win in 1980.
Oops I better get to sleep

, I try to work on this later

, this is just my ideas so far. Yes I know you democrats are probably pulling your hair out, 8 years of Reagan/Romney followed by 8 years of Romney/Ford. The good news is almost no party holds onto the seat more than 20 years historically at a time.