There certainly isn’t a clear class probability or model or any truly objective prediction method of nuclear war and the nature, extent, and cost of its damage. But generally three things can happen - civilization is destroyed temporarily, permanently, or not at all. If it is permanently destroyed, prior contracts don’t really matter anymore - there would be no legal system to enforce them, in addition to the likelihood that the insurance agency and its assets are long dead and gone. In the other cases, you find an agency to enforce the contracts, provided the insurance company is still around. I’m sure many insurers would go bankrupt in such a scenario; however, this is simply because this is a risky and cutting-edge business.
One thing that should be pointed out is if insurers operated in such a capacity, they would have an incentive to prevent nuclear war, using everything from lobbying to spy networks to avoid liabilities.
While insurance is about risk prediction and cost estimates, these things can be wildly wrong and the company can still be solvent so long as it takes in more money or holds more assets than it is required to pay out. So while the risk is very difficult to assess, this can be overcome by high premiums.
As far as uniform law, there is no uniform law, as in everybody can’t do these things. You should read David Friedman’s Machinery of Freedom to get a better picture of the operation. Basically, private courts would each uphold their own ideals of what the law should be. Private insurers/defense agencies would designate which courts it would use in which cases in its contract to the end customer.
So let’s take the issue of capital punishment. Let’s say Bob is for it, but Carl isn’t. Bob hires a defense agency that supports capital punishment, but Carl hires one that does not. Carl kills Bob, and Bob’s defense agency attempts to arrest him. However, Carl’s agency has an obligation to protect Carl. The agencies could battle, but that’s costly. So instead, the agencies will agree to limited cooperation. If agency A wants to arrest agency B’s client, they will allow it, so long as the client is tried in a court of agency B’s approval. In the above case, Carl’s agency will obviously allow him to be arrested and tried for murder, as long as he is tried in a court that does not support capital punishment. But Bob’s agency wants him to be tried in a capital punishment court. It comes down to money. The agencies will make a deal. One of the two agencies will have to buckle on the issue, and agree to support a court whose ideals of law vary from their own; but in exchange they will receive compensation. If everything else is equal, the defense agency who defers which courts their clients can go before peacefully can offer cheaper premiums to their clients, although the quality of their service may be considered lower. Thus, consumer preferences will determine how things usually work out. But look how well this reduces conflict - let’s say Carl kills Dave, also against capital punishment. Dave’s defense agency will send the case before a court that does not endorse capital punishment, in cooperation with Carl’s. Even if 99% of the population is against ____, the 1% that is for it can do it without harassment, so long as they do so in fellow company or they pay exorbitantly high defense agency premiums.
I am only skimming the article mentioned, but it seems to amount to “what if criminals refuse to obey the order described?” I don’t think that’s a reasonable argument - one could make the same of any system. Why is a centralized, aggressively-coercive state more competent at preventing or punishing crime AND upholding civil rights than a decentralized system of voluntary private enterprise? If this is not the crux of the argument, the argument is simply grasping at hypothetical straws.