OK. Feb '13 Senate passes its version of a constitutional amendment changing the term of the presidency. House debates it, passes its own version in ?April?. The two versions get reconciled in ?May?. You now have to get a supermajority of states to ratify it. 1) how likely is this, at all, and 2) can it possibly be done fast enough to change the set-up of the presidential election?
I suspect strongly that the '16 election goes ahead as planned and the '20 election is the first one under the new rules IF the amendment passes.
Actually, the chances are quite good on both counts, due to the special circumstances of the time.
The Democrats, whose Amendment this is [1], comfortably control the HoR and most of the State Legislatures. They are shy of two-thirds in the outgoing House, but well over it in the new. The Republicans, for their part, are split right down the middle, with Taftites supporting the Amendment (as it will eliminate TR from the Presidential stakes [2]) while Roosevelt Reps, plus of course the handful of Progressives, oppose it.
In such circs, the House vote looks like a pure formality, and if the voting pattern in the Statehouses mirrors that in Congress, so does ratification. There are probably a few Western States where Progressives and TR Republicans have control on their own, and will therefore reject, but nowhere near the 13 required to block it. It could easily be in effect by the end of 1914.
Things get a lot more iffy in the case of an incoming President Bryan trying to revive it in 1914/15. By then much of the heat has gone out of the issue, with the Progressive Party clearly a busted flush, not requiring such drastic measures, and TR having no hope of the Republican nomination. In those circs, it might indeed not be ratified in time for 1916, and maybe not at all.
[1] OTL only one Democrat voted aginst it in the Senate, though several abstained or were absent. The Republicans and Progressives (one or two Senators were elected on both tickets, so the distinction is blurred) were almost evenly divided, with 19 voting yea and 22 nay.
[2] It will of course eliminate Taft as well, but he is out of the game anyway.