As far as I know, Franklin Delano Roosevelt was our first President frequently referred to by his initials, as in FDR. I assume this came about due FDR's long last name having 3 syllables.
I know some people, like Richard Nixon, refer to FDR's cousin Teddy as TR, but I don't know if Teddy Roosevelt's contempories did.
FDR's successor, Harry S Truman, did not become known as HST. The S between Harry and Truman is not the first letter of his middle name. Truman's middle name is a middle initial. With no period after it. Why, I do not know.
Truman was followed by a President with a long last name, but he did not become known as DDE. Instead he was known as Ike. Ike was Dwight David Eisenhower. Ike sounds better than saying DDE.
Ike was followed by JFK. Who was followed by LBJ. LBJ had a fairly short last name. But I think people liked the sound of saying JFK and so they segued easily into Lyndon Baines Johnson being LBJ. It worked great for anti-war chants, as in "Hey Hey LBJ. How many kids did you kill today?"
LBJ was to be our last President known by his initials. Nixon followed him and while there were some instances of him being referred to as RMN, it just did not stick. Mostly, I suppose, because Nixon is a nice short name with a punch to it when said aloud, like Hitler.
Nixon was followed by Ford. Again a short name. Then Carter. Again short. I think Jimmy Carter's middle name is Earl. That'd make him JEC. That just looks weird. Jimmy Carter was followed by Ronald Reagan. I do remember seeing Reagan referred to as RR a time or two, but that definitely did not stick. It was just way to easy to say Reagan, a good short name, like Nixon and Hitler.
Reagan was followed by Bush. No need to call him GHWB. Bush was followed by Clinton. Again a nice 2 syllable name that has a punch to it, so it was Clinton, not WJC. That would have looked to much like the initials for Water Closet.
Of course, Clinton was unfortunately followed by another Bush. Who was never referred to as GWB, but sometimes the, to be retired tomorrow, Bush was referred to as W.
Tomorrow the world breathes a sigh of relief as W is replaced by Barack Hussein Obama. I'm fairly certain he will not be referred to by his initials. BHO sounds too much like HBO. And if you take the middle name out you are left with BO. And that definitely would not sound Presidential.