I read recently that the average human carries around 3 or 4 recessive lethal alleles, genes for a fatal disease that are not expressed because there is another copy of that gene that does not cause disease and suppresses the lethal one. If someone (necessarily an embryo or fetus) has two copies of the lethal gene, though, the lethal characteristic will be expressed and will result in either a miscarriage or very early death.
So, instead of creating science-fiction mutants, inbreeding simply causes a higher rate of miscarriages and infant mortality. (Many, if not most, harmful mutations are sufficiently harmful to be lethal.) The reason for this is that the 3-4 lethal alleles each human has are likely to be for entirely different genes, while two closely related individuals may carry lethal alleles for the same gene. For each, there is a 50% chance that the lethal allele will be passed on, and a 50% chance that the healthy dominant allele will be passed on. Thus, if two individuals mate who carry lethal alleles for the same gene, 25% of the offspring will have two copies of the lethal allele and will not survive.
If you were the last man on Earth (and you carried 4 recessive lethal alleles) and you mated with the last 20 women on earth, there would be a 50% chance of passing on each of your lethal alleles to your offspring. There wouldn’t be any effect in the first generation, but later generations would suffer enormous infant mortality. (Some of your offspring would even have three or all four of your lethal alleles, and most of their offspring would die.) Note that I’m only considering alleles which are outright lethal and not those for genetic diseases.
Because of this, it would be much better if there were 10 last men on earth and 10 last women on earth rather than 1 last man and 20 last women. The birth rates would be higher with 20 women, but they would only be doubled, and the higher rate of infant mortality might cancel this out.
Another factor is that women under stress ovulate less frequently and have miscarriages more often under stressful conditions. Presumably, if some disaster happened that reduced the human population to around 20, the survivors would be under enormous stress.
One final issue to consider: After the first generation, female offspring would be unable to mate with anyone with whom they were not somehow genetically related. For the males, the 19 women who were not their mothers or half-sisters wouldn’t be genetically related, though they would technically be in-laws and still very disturbing. Genetically, though, if the 19 other Last Women hadn’t reached menopause before their male offspring reached maturity, that would be the choice of least inbreeding.
Opinion part: If I was the Last Man On Earth, I think that’s the end of humanity, because the genetic/moral consequences of it are too disturbing to be worth saving the species. Hopefully, if there is ever a Last Man On Earth, he won’t have taken a course in genetics, law, or religion.