Will the engine have any significant damage?

Possibly, possibly not. This is the reason they're called idiot lights. If you see a red one with a little picture of a battery come on, but choose to ignore it and keep driving until something dramatic happens, then glance at the temperature gauge and say 'gosh wow!', then the name is quite apt.

There are ways to test for major damage (visually inspecting the coolant and engine oil, pressure/leak testing the cooling system, testing for the presence of hydrocarbons e.t..c.), but I'm guessing your friend will just ignore anything else the shop says needs doing after replacing the belt. Then yet another of 'those' questions will be posted here.

I don't get how some peoples minds work. Do you think anyone that has ever learned anything about engines is magically linked to every engine on the planet, and can tell it's exact status by looking into a crystal ball or something? Hahaha! It's just bizarre.

Seeing that the car is at the shop as stated in your comment what did you expect to learn here. The mechanic, unless tearing the engine all the way down will start the car and check for anything odd like water in the oil but if nothing is found he will tell your friend it is alright and send them on there way. Is there damage? Your friend won't know because no one will pay for the engine to be torn down to find out if the piston walls are scorned and the pistons are damaged so what does it matter. Your friend would have been better off to shut the engine down as soon as they realized the belt broke buy I know it was in a bad neighborhood and it was not safe to stop or some other stupid reason so what does it matter.

The instant the belt broke, the alternator light was on, or the voltmeter showed a big drop. At the same instant, the water pump stopped circulating coolant. That was the right moment to pull off the road and shut off the engine.

But, the car kept going, and the engine block and cylinder head were cooking.

It doesn't matter WHAT the pace was. Overheating can warp an aluminum cylinder head, then the head gasket fails. There could be cracks in the area of the exhaust valve seats (in the cylinder head).

Answer: Yes, it could.

Diving with an overheated engine can cause the cylinder head to warp and crack, also the cylinders can get galled by over-expanded pistons. Repairing this much damage can require disassembling the engine, honing the cylinders, replacing the pistons and rings, repairing the cylinder head if it can be repaired. If the shight fairy was on vacation, maybe all these problems didn't come to pass, but you described the best way to do all that. Replace the belt and see how it runs, but make sure the cooling system is refilled. Good luck.

The sensible thing would to have stopped when the belt broke and the charge light came on.

Letting it overheat was not a good idea. You may have to have major engine work done.

Having proper servicing should have replaced the belt before it broke.

Over heating can do many things such as warp the head , drop the viscosity of the oil resulting in scored cylinders or damaged crankshaft bearings. Odds are it didn't do any real damage but look for signs of antifreeze in the oil.

With the belt broken, the water pump isn't circulating coolant. So yes it could have done all sorts of damage, no matter what the temperature gauge said. No water circulation means some parts of the engine were probably much hotter than the gauge suggested.

I would say Very Possibly. On the other hand I have had cars not only overheat but also get very loud before I could get off the road. And yet, I never did any damage that I knew about.

Hi the serpentine belt moves the water pump and alternator so she should stop immediately other wise the car will be damaged.

It should be okay, if she shut it down when it got hot, and didn't run it for awhile. Only way to really know would to be replace the serpentine, and let it idle for awhile. Listen for any strange noises, such as clicks, taps, or chirps. More than likely, you'll be fine.