How did bombing Safin's island fix the nano-bot problem?

Nanobots are not indestructible. The problem is that their human host is not indestructible, either.

In most sci-fi, nanobots are microscopic machines capable of self-replication. I think No Time To Die suggests that these 'bots are living micro-organisms based on the smallpox virus. After all, if Heracles were tiny robots, then Bond's EMP watch would permanently disable them (not to mention his own earpiece, but I guess electromagnetic radiation works differently when you're an arrogant writer who thinks people at parties will be impressed that you had the temerity to kill a beloved protagonist).

If that's accurate, it means that Heracles can probably be killed with bleach or some other disinfecting chemical, or maybe an autoclave would be needed (very high temp & pressure). But bleach will also kill a human, and I suspect an autoclave would reduce a human to a hot, chunky soup. Cancer is basically the same: cancer is really easy to kill, but not if you want the patient to live.

Whatever the specifics, Heracles is only "permanent" in the same sense that AIDS is: it's not magically invulnerable, but it can't be removed from the body, and anything that might kill it in situ would also kill the human.

Since Heracles is based on smallpox, it likely can only infect things that smallpox can infect, which I imagine means it can only infect homo sapiens and maybe a few closely related species of primate. Thus, it won't escape the island by infecting sea life. And I think many virii cannot survive for long outside a host, so any free-floating particle will die before the wind can even carry it off the island. Nor is ocean water an especially hospitable environment for an organism that is designed for life inside a warm human body. A weaponized version is probably hardier, but there are limits to what could reasonably be achieved.

Sterilizing the island via carpet-bombing will probably get almost everything. To be safe, they'd want to send in HAZMAT teams later to meticulously sterilize everything again.

I will note that there is at least one kind of naturally-occurring microorganism that is essentially invincible: the prion, made famous by Mad Cow Disease. They might not be "alive" in the conventional sense, but, just like a fictional nanobot, a single prion can convert a healthy protein into another prion, and we really have no way of truly killing them. That's because what makes a prion dangerous is its physical shape: it's basically a dumb mechanical contraption that reshapes other contraptions to look like it does -- turning them into prions like itself.

The nanobots can be programmed to attack certain genetic markers. They will only harm people with those markers. Everyone else is unaffected.

All the nanobots in the base were unprogrammed, with the exception of the ones directly targeted at Madeleine and Mathilde. Scattering them into the ocean doesn't matter. Even if they spread around the world, it doesn't matter. They won't harm anyone. And Bond made sure that the ones for Madeleine and Mathilde were incinerated.