So, to tie the dates down, the Senate gets cloture on the amendment, and it passes in 1970; since it is unlikely to affect Nixon's re-election* (to start, it may not be ratified in time for 72), we're looking at 1976 being the first affected election, which OTL was very close where the popular vote was concerned. If Watergate still happens and Ford still ends up the unelected incumbent, then our first big question is does the expectation of a different game affect the Democratic Primaries?
*that said, if your PoD is Nixon pressuring small state Republicans to let it come up to a vote, that would have butterflies in 72 -- not enough for Nixon to lose, mind you, but still...