Both could well have lost, but that doesn't mean that they don't end up running for other positions later on.
But assuming a Nixon loss in his 1950 run for the Senate, he'll also be out of the House, for the time being. Warren won't appoint him to the Senate upon the resignation of Vice President-elect Bill Knowland in 1953, appointing (as IOTL), Thomas Kuchel. With Knowland out of the Senate, Kuchel and Helen Douglas representing California and Earl Warren (until October '53 anyway, when he is replaced by fellow liberal Goodwin Knight) in the Governor's Mansion, the conservative Nixon has little wiggle room to move up.
He wins re-election as a member of the House in 1954 and decides to challenge Helen Douglas in a rematch in 1956, which he loses in spite of narrowing the gap between the two. Deciding in 1958 (with the backing of Vice President Knowland) to challenge Knight rather than take on Kuchel, Nixon runs for Governor. It's a wash for the Democrats, as with a right-to-work ballot measure (strongly favored by candidate Nixon) knocks out Nixon (who does defeat Knight in the Republican primary) who loses to Pat Brown, and Kuchel, who loses to Ronald Reagan, the former actor turned Democratic Congressman and darling of the social democratic left.
At that point, Nixon's career is viewed as essentially over. Out of the House and having been defeated in a bid for the Senate and the Governorship, he resigns himself to private life. In 1965, he ends up with a cushy post as Chancellor of the University of California at Berkeley, presiding over a raucous campus environment that gives birth to the New Left. Nixon, harkening back to his old days in Congress as a "red hunter" takes his opportunity as President of UC Berkeley to wage a war on student radicals and makes enough hay of the issue to revive his political career. In 1966, Nixon successfully avenges his 1958 defeat at the hands of Pat Brown, becoming Governor of California on the Republican line.
Nixon's "rise from the dead" is seen as a comeback for the red-baiting kind of conservatism he popularized during the early Cold War period. Headlines across the country name Nixon the "man to beat" in '68, although incumbent Henry Jackson, elected in 1964, still has relatively high approval ratings. Nevertheless, Nixon stands as a candidate in the '68 contest, losing the Republican nomination to Michigan Governor George Romney, who subsequently loses the White House to Jackson.
Re-elected in 1970, Nixon launches an early bid for the White House in 1971 and by mid-1972 has the Republican nomination under his belt. For a running-mate, Nixon chooses Texas Senator George H.W. Bush as a sop to the party's moderates. Nixon and Bush win a close race against the Democratic vice president, making Nixon the nation's 37th President come January 20, 1973.
Nixon will serve until January 20, 1981. His administration oversaw a general expansion of social welfare benefits, a withdrawal from American operations in the Congo, and detente with the People's Republic of China, as well as better relations with the Soviet Union.
I'll do a Kennedy write up later.