It's beyond my ken to comment on the universality of cord cutting. Going back to the Hippocrates, however, we can see advice on cutting the umbilical cord in the Western medical canon. On the other side of the Atlantic, in Mesoamerica, we also see the practice of cutting the umbilical cord. Seeing as how Hippocrates and the ticitl of the Aztecs were separated not only by an ocean, but also a pattern of human migration that placed them tens of thousands of years apart, we can at least say the practice was widespread.
This is not to say the understanding of the physiology or cultural significance of the umbilical cord was identical across time and place. Hippocrates seems to think the cord pulsations represented air, which is actually part of longstanding misconception in the Hippocratic tradition of thinking the arteries carried air, not blood.
The Aztecs do not appear to have the same misconception about the vascular system (as befits a culture that was not shy about dismemberment). The umbilical cord, however, was seen as a link to the child's destiny. As such, after it was cut, the cord was preserved to be buried in a placed as befitted the fate of the child. For a girl, it would be buried in the home, next to the hearth, signifying the domestic sphere assigned to women in Aztec society. As the midwife did this she would recite a sort of prayer, which went, in part:
Thou wilt be in the heart of the home, thou wilt go nowhere, though wilt nowhere become a wanderer, thou becomest the banked fire, the hearth stones. Here our lord planteth thee, buried thee. And thou wilt become fatigued, thou wilt become tired; thou art to provide water, to grind maize, to drudge; thou art to sweat by the ashes, the hearth.
For a boy, the cord would be passed along to distinguished warrior, to be buried on a battlefield, symbolizing the ideal role of a soldier for a boy. As with the girls, the midwife would intone words of destiny as she cut the cord, saying:
I take, I cut [the umbilical cord] from thy side, from thy middle. Heed, hearken: thy home is not here, for thou art an eagle, thou art an ocelot...
Eagles and ocelots were, of course, two military orders with in Aztec society. As with the dire warning to the girls about drudgery and sweat, however, the boys were also reminded of the inevitable downsides of even an ideal life. For the midwife continued:
Thou hast been sent into warfare. War is thy desert, thy task. Thou shalt give drink, nourishment, food to the sun, the lord of the earth. Thy real home, thy property, thy lot is the home of the sun there in the heavens. Thou art to praise, to gladden Totonametl in mania. Perhaps thou wilt receive the gift, perhaps thou wilt merit death by the obsidian knife, the flowered death by the obsidian knife.
Of course, all of this dire destiny foretold would have to wait for the child to grow up. The more immediate concerns were to bathe the child and arrange all the rest of the feasting and speeches which accompanying a new birth, culminating in the naming of the newborn.
All quotes from Sahagún's General History of the Things of New Spain, Book 6: Rhetoric and Moral Philosophy, 1969 trans. Anderson & Dibble.