If the engine had to be cranking to get a fuel flow it must have a fuel pump. Please post the Model, Type and Code so we can all get on the same page.
A silly question: are you sure the substitute carburetor was OK? It hadn't been out of service, resulting in it being gunked up somewhere?
Because it will run if you prime it directly into the cylinder, the problem isn't ignition or compression - it is fuel. Do you have a massive vacuum leak somewhere - e.g. carburetor insulator or gaskets incorrectly assembled? Vacuum tapping left open?
It sounds as if we need a bunch of pictures of the engine in the mower, and try to do this from first principles the way we normally do here. I know this doesn't meet your needs in this case because of the work situation.
