Hand-and-a-half and two-handed swords never replaced single-handed swords. They were always a specialty weapon that existed side-by-side with standard single-handed weapons. In an evolutionary sense, they were a branch, and a somewhat short-lived one, running from about 1300 to about 1550. They were impressive weapons, so they did in fact persist for a while longer, but mostly as ceremonial weapons. By the late 16th Century they were pretty much done on the battlefield.

If you take the early 1500s--peak-two-handed-sword--as an example, typical infantry swords were actually single-handed weapons like the katsbalger, falchion, and various proto-rapiers (single-handed, straight-bladed cut-and-thrust swords with early developed hilts). For many soldiers, these were backup weapons, with some sort of pole weapon such as a pike or halberd being primary. If you used a two-handed sword in 1550, you were a specialist, probably an infantryman trained to go after pike formations, and probably getting extra pay for it.

At the other end of the two-handed sword era, say late 13th or early 14th Century, it's a very different story. Swords at this time were primarily cutting weapons, and shields were the main defence. The trend to longer swords (eg. Oakeshott type XI) was driven by horsemen/knights who had a requirement for longer reach, whether striking down at footmen, or reaching for other horsemen. These had to be single-handed weapons, because you didn't use both hands to swing a weapon from the saddle. Any handling awkwardness from not being able to put your full body into swinging a longer weapon was made up for by the energy of the charging horse. But once dismounted, these weapons became awkward and underpowered. Solution: longer grips for two handed use when dismounted, leading to hand-and-a-half cutting swords of types XII and XIII.

In between these two extremes there was a fair bit of evolution. The knightly hand-and-a-half sword evolved into a steep-pointed thrusting weapon to cope with heavier armour in the 14th and 15th Century, which necessitated wider spear-like grips, even involving placing the 2nd hand onto the blade itself. The adaptation of pole-fighting techniques to the sword made them of increasing interest to pole-arm-equipped infantry, which then drove the evolution in another direction, eventually leading to the 16th Century zweihander, a pure infantry weapon for use against pole weapon formations, as noted above.

It is worth noting that 16th Century two-handed sword specialists often carried a regular single-handed sword with them as well--you would need it if an actual sword fight broke out! This helps to understand how specialized and niche these weapons were.