It’s all about the headlines.
It started with FDR. “Roosevelt” is nine characters*. It took up a lot of space, so headline writers started with the abbreviation. The presidents before him – Harding, Coolidge, and Hoover were all shorter, and headlines were more formal, so abbreviation seemed disrespectful. “FDR” came about as things became less formal due to the Depression, and it stuck – it’s only three letters, which gave headline writers more space (and one or two character can make a big difference when formulating a headline).
When Truman came along, it’s six letters, so the abbreviation wasn’t necessary. Then came Eisenhower, for which “Ike” became standard.** “Kennedy” was seven letters and the idea of a nickname became entrenched, so “JFK” is was. Same for Johnson. “Nixon,” though, was only five letters (and the “i” was half width), so there was no need. Then came a bunch of short presidential names: Ford, Carter, Reagan, Bush, Clinton, Bush, Obama. None are more than six letters (and “Clinton” had two thin letters, making it five letters for headline use). So the practice has died out. The candidates this year – Clinton, Sanders, Trump, and Cruz – all have short names, so I doubt the tradition will be revived.
As for HST, I doubt it was regularly used for Truman until recently. The nGram viewer shows no jump in the term at the time he became president, which is clear when looking at FDR and JFK. It’s probably the same for TR, though nGram doesn’t help, since the term was used to abbreviate “translate.” But headlines were wordier during his presidency, so the longer name wasn’t a problem.
As a side note, the New York Times refused to use these nicknames. They had a special font made for “Eisenhower” to help it fit into their headlines.
*Actually, the values for the various letters were different in headline writing of the time, but I don’t know all the values. It was something like lower case being 1, letters like “i” being 1/2, letters like “w” being 1 1/2, upper case 1 1/2 - 3, etc.
**Eisenhower didn’t like that nickname at all.