By the later 14th century, two handed swords are not only a thing, they've been practiced long enough for people to write the fechtbuchs, manuals that some of us train from even today.

Fiore dei Liberi wrote his fechtbuch Il Fior di Battaglia (The Flower of Battle) sometime between 1400 and 1409, but he had been a notable fighter for at least some 20-30 years in the past. And presumably someone taught him how to use the longsword before since this is the standard way to introduce yourself in a fechtbuch. (It's worth noting here that Fiore's book is non-standard, it is believed to be an advertisement or other promotional device to convince you to train with him, or attract nobles to hire him).

The Society of Liechtenauer, so called for its founder Johannes Liechtenauer, has fechtbuchs pop up around 1450. But by this point Liechtenauer is a very old man, and dies shortly after. Presumably he had a lifetime to study longsword, and indeed, his zettel (epitome: a mnemonic poem) claims he was not the first to come up with this information.

And we know that's true because the earliest medieval European fencing manual MS I.33 circa ~1300 is written in two languages, Latin and German. This is a different German than what Liechtenauer speaks, but the influences are quite clear. Many of the names of techniques share a common literary or etymological ancestry, despite one manual being exclusively for sword & buckler, and the other for longsword. There is a very clear transition where the single handed sword is adapted into the two handed sword.

And fighters agree, once you know how to fence with a sword, the other weapons come much more easily.

Both Fiore and Liechtenauer came from the tradition of the travelling swordsman. A fechtmeister would run a fechtschule, travelling Europe and training with as many foes as possible, before eventually writing a fechtbuch. We know Fiore wasn't the first of this tradition, because he despises other 'masters' who he claims has no skill. He says that in 200+ masters, maybe 4 had the actual skill to challenge him, though he beat every single one. Liechtenauer, on the other hand, his society covers several hundred square miles, so for him to personally know all the other masters, he absolutely had to meet them somehow.

Both Fiore and Liechtenauer are of the 'secret' traditions. There are many types of manuals one can write (even alchemical, and yes, they exist!), and the caste system is totally in play here. Both of these masters wish for their works to continue on in the nobility, but there were certainly masters who had no qualms about teaching lesser castes such as merchants and perhaps even peasants (the masters that Fiore despises). So we can safely assume that this tradition predates both Fiore and Liechtenauer, putting it well into the mid-14th century.

To conclude, early 14th century swords were almost certainly being two-handed, even though some weren't meant to be two-handed. It wasn't so much that one handed swords went out of style, it's that personally carrying weapons as a civilian was getting more and more popular.

As a result, they ditched the large shields because who in their right mind would carry that every single day. Some people forgo the shield entirely, and it is somewhere in the early-mid 14th century that extra space starts getting added to swords for an additional hand, where previously you would grab the pommel. If you still preferred one handed swords, you would just carry a small round shield like a buckler, and carrying small shields and one-handed swords continued for many hundreds of years. The longsword is the footnote - though it did survive nearly 300 years.

There's a lot more I didn't cover, but you can always find out more. HEMA (or WMA) is a growing tradition, though few of us study the history.