Bobby did not have the charisma, particularly in 1960.
Since Jack did not do a lot in the 1950s, it seems plausible the butterflies would not be too large. Therefore, I would suggest the 1960 candidates would be similar to those IOTL. Since LBJ was the runner-up in the nomination, it seems most likely he would have been the candidate. An LBJ-Nixon race seems particularly interesting. Given LBJ's ruthlessness, it seems plausible that he would win. It seems likely Johnson would pick up a few Southern States (e.g. TN, VA, and MS (which had "unpledged electors on the ballot). Unless, he loses big Northeastern states, LBJ would win easily. LBJ becoming President in 1960 creates some very interesting possibilities. I would say it is almost certain the Soviets are less aggressive (e.g. no Cuban Missile crisis, perhaps no Berlin wall). The truly interesting question becomes, would LBJ pursue his Great Society and Civil Rights agenda with the same vigor he did IOTL. I think the answer is yes on the first and probably on the second.
A more interesting question is whether the lack of JFK would allow another candidate (such as Humphrey) who never really took off IOTL to do well enough in the primaries to win on a second or third ballot as the alternative to the rather crude wheeler-dealer Johnson. My intuition is Humphrey loses the entire South (either to Nixon or to an unpledged slate) and probably Illinois, resulting in Nixon in 1960.