I was listening to Dan Carlin's Hardcore History podcast and when I heard him say this I was awestruck. I did a little bit of reading on it, but I couldn't find much info, just the same story regurgitated over. I then thought maybe it was a legend that either the Mongols or even Khan himself made up to make themselves seem even MORE intimidating (as if they needed to). Or perhaps it was a legend/tale passed down from travelers, people he conquered, people who feared him, his enemies - whoever...
But I wonder, is there any evidence that such an event took place?
It's true that the Mongols razed Nishapur down to the ground, and before that it was a big city. The Mongols razed down other big cities too, such as Samarkand, Herat, Bukhara etc. but they all were rebuilt under Mongol rule in a short time, only Nishapur never saw its former glory again, which allows us to make assumptions on level of destruction in Nishapur.
Razing down entire cities was a Mongol psychological warfare tactic, as mentioned above: "If you surrender with your own consent, you will live your lives as free citizens of the Mongol Empire and we'll protect you as long as you keep paying your taxes. If you don't, then we'll butcher you and everyone you know, take your women and children as slaves, and burn your city down to ground". Previous Turkic-Mongolic steppe empires/states also followed this rule (a sieging army asks the besieged to surrender three times. If the besieged refuses, then in case of a victory, lives and property of the besieged is at sieging army's disposal, which means a sack of the city), however none of them used it as effectively as Mongols as a "siege instrument". Anyway, Nishapur was one of those first cities which refused to surrender.
However, there are several issues:
From first hand sources, we know that the Mongols created such a psychological 'despair' environment that the chroniclers/observers write it down with surprise that "a single Mongol warrior orders more than ten able and strong men to follow him, they do it without any resistance, and the Mongol warrior kills them one by one when he sees fit, again without any resistance". So we're certain that the Mongols slaughtered a lot of people in Nishapur (there is a famous urban legend/story which Arab historians tell: the Mongols were killing the civillians by shooting arrows, but there were too many to kill that they tied 2-3 people together, took them to a bridge on a river and shot only one arrow to one person and let others drown. Although story might not be real, it gives an idea about the slaughter). However, it's virtually impossible to kill "entire population" of such a big city, even for Mongols, let alone killing by "beheading" them all.
Secondly, I'm not an expert, but I can say it's also impossible for Nishapur to be inhabited by 1.75 million people considering the era. Two biggest cities of the world prior to Mongols were pre-crusade Constantinople and Baghdad, and both had at most 500.000 to 750.000 people, which also is debatable.
Lastly, such inflated numbers are result of Arab/Muslim chroniclers'/historians' hatred for Mongol conquests (as the dominant military power of their time, achieved great success in science and literature, built a great empire and a widespread religion, Arabs/Muslims were rather proud and arrogant at the time. The humiliation they faced before the Tengriist Mongols were no less than the humiliation Sassanids faced before the bedouin Arabs). Most of the time, especially recording the atrocities commited by Mongols in Muslim inhabited lands, they tend to inflate the numbers of people killed/enslaved, and exaggerate the level of cruelty commited by the invading Mongol armies due to this very reason. So even if there's a "1.75 million" number written down somewhere in a first hand source, it most probably is a result of this habit.
Grisly as it is, victorious armies putting cities to the torch for much of history isn't unheard of. Empires like the Assyrians exacted overwhelmingly brutal retribution on rebellious cities or territories to scare others into submission, and even groups as late as the Wermacht in WWII would use mass reprisal to try and crush dissent, in the Mongols case many think that by spreading such terror of them more cities would surrender without a fight. So the idea of the Mongols butchering a city isn't outlandish in itself, although maybe the numbers were inflated by a degree -killing an entire population that large by hand is extremely difficult even for a group as organized and efficient as the Mongols.
The numbers could be rather high of course, but it's perfectly within reason to believe the Khan did have his army slaughter an enormous number of civilians in the city.