Much faster than you might think. One generation!
For this story, you'll have to dump those silly robots. Instead, imagine this. Some years in the future, doctors become aware that a rapidly increasing number of young couples are attending fertility clinics. Plenty of young couples are still giving birth to healthy babies, but far more than in previous generations, can't.
The penny drops. It is discovered that these couples are infertile because one of them has an odd number of chromosomes. He, or she, is sterile for the same reason that a mule is sterile. And of course, this means that part of humanity has evolved, un-noticed, very suddenly, into a new species with two more, or two less, chromosomes. You can't tell them apart, other than using a microscope to count chromosomes. Or more likely, you can hardly tell them apart. Once you know what you are looking for, you can tell, though with rather less than 100% reliability absent scientific confirmation.
Over to you, the author. I've no doubt that the new homo species will have inherited the same amount of bigotry and boneheaded stupidity as the original. It might be interesting to make the narrator or hero one of those sterile hybrid children.
OK, to the hard science. Consider the horse and the donkey. Two separate species sharing a common ancestor. Yet a donkey has 62 chromosomes, and a horse, 64. How can they share a common ancestor? How can any creature have a different number of chromosome pairs to its parent, and yet manage to breed non-sterile offspring?
If a single proto-donkey had once given birth to a single mutated foal with an extra chromosome pair, that would be a genetic dead end. The only way that foal could reproduce is if, in the same generation, a significant fraction of other proto-donkeys were also giving birth to proto-horses. To repeat: unless something triggers the same genetic repackaging in a significant percentage of a species within the same generation, a new species with one more or one fewer chromosome pairs cannot arise.
It is believed that a widespread virus or retrovirus infection is the probable triggering mechanism.
I've talked of horses and donkeys because mules are well-known. In many ways they are superior to both their parents. But they are sterile. Something similar must have happened in the very recent evolutionary history of homo sapiens. We have a different number of chromosome pairs to our nearest extant related species, the chimpanzee. The causative virus may still be out there. Or if it's a retrovirus, it may still be latent within us.