The conversion factor for nitrogen from crude protein is 6.25. So if you have hay that has 12% protein. A ton of that hay has 240 pounds of protein. 240 divided by 6.25 is 38.4 pounds of nitrogen per ton. The nitrogen that passes through a cow ends up about half in the form of organic nitrogen and half in the form of ammonia. There is very very little nitrate in fresh cow manure or in the urine, way less than 1%. Nitrate is the water soluble form of nitrogen which can and will leave easily. It is also the form of nitrogen most readily used by the plants. The organic nitrogen isn't going to go anywhere. It takes time and the right conditions for it to mineralize into plant available. The ammonia is the part than can leave but might not. It can convert to ammonium and volatize off into the air. Some of it will. It can convert to nitrate and leach off. It can also be used by the plants. There are way too many variables to guess at how it will act after passing through the cow.