Cranks but will not start

Normally a bad fuel pump isn't going to make a loud noise it will just stop turning and you get no fuel pressure. First thing I would do as it made a noise would be to check the compression, It's an easy process and you tell you if the noise was internal engine damage like the timing chain or something else has failed.
https://www.2carpros.com/articles/how-to-test-engine-compression
Another quick thing would be to remove the oil fill and see if you can watch the camshaft when someone tries to start the engine. Another test would be to spray some carburetor cleaner into the air intake and see if it fires on that. That would at least show you if it has spark.

How do I check compression if the engine only cranks?

That is all it needs to do. You disconnect the wiring to the ignition, remove all of the spark plugs, block the throttle open and then install the compression gauge. Now turn the engine over until the compression gauge stops rising and write down that pressure. Then move to the next cylinder and repeat the process.
When you have all 4 pressures then you can compare them. Say you find the first cylinder has 140 psi, the next 130, the next 60 and the last one has 30. That would not be a good set of numbers. Ideally you want them all within 10 percent of each other from high to low. So if the high pressure was 120 you want the lowest to be no lower than about 110 or so with the other 2 falling in that range.

I've taken the compression of the engine. In the first cylinder there was no compression. In the third there was weak compression. The second and fourth were fine.

That isn't a good sign. I suspect that the number one piston isn't moving, the loud noise may have been the connecting rod breaking. To check that you may be able to use something flexible like string trimmer line or a piece of wire and put it into the spark plug hole and see if it moves if you crank the engine over by hand. No motion would mean there was a major part failure and you need a replacement engine.

If it does move, what would happen then?

Then you would need to pull the valve cover and try to determine what failed. Or use a camera to look inside the engine and see if you can tell why there is no compression.