88 F250 4.9 in-line6 NO-START
 

On the older engines we usually needed fuel, air, spark and correct timing and the engine usually started. If the intake is clear and filter is reasonably clean the air consideration is met. Spark can be confirmed by removing one of the spark plug wires from the plug, inserting a small screwdriver into the wire terminal end , placing the screwdriver shaft approx 1/4-3/8 in away from any metal part of the engine and cranking the engine over . You should see a spark going from the screwdriver shaft to the engine ground (metal). If you were to remove one of the spark plugs after trying to start the engine ,it should be pretty dry, if wet it is overfueled, flooded. Holdong the throttle pedal to the floor when cranking will prevent addl fuel from getting into the cyls and result in what is referred to as clear flood condition.
If all the plugs are dry it could be due to no fuel being put into the cyls.
Suggest you check the above 3 checks before going further and let us know what you find.

Since the "Battery to Ground" cable was replaced, was the body ground also connected and does it have a good body ground? Will it fire up if you spray fuel down its throat?
Reason I ask is because the CPU uses the body ground to get its required multiple grounds. The CPU fires the injectors by grounding the injectors. No ground and the injectors will not fire.

Get your ignition in order first. Plugs, wires, distributor, coil, etc. it sounds like it's really messed up. There is an ignition module on the steering column and that is the usual problem with Ford's of that year. That may be why the previous owners went to push button start trying to avoid that problem. Many of the engines electronics goes through that module.

Not sure if that engine had the older distributor with points .If so let us know. If so we can assist you on checking the points .

If it is of any help,
This vehicle should have Electronic fuel injection, which means it requires a signal to allow the computer to maintain the ground for the fuel pump relay. The EEC relay gets energized when the key is turned on and that relay closes and provides the circuit for the fuel pump relay. The signal we are looking for comes from a pick - up coil in the dist and goes to the ECA which now controls the ign module that provides the spark control to the coil and rotor to the plugs. Somewhere we are either losing the signal or it is not being generated at all. The object is to do some checks to determine what is happening and where.Some of the things we need to confirm are, is the EEC relay working, is the fuel pump running for the usual 2seconds when the key is first turne on? Since we already know there is no spark, it might be a good idea to confine checks to the ign system. Don't know equip you have to work with , so can you fill us in there?

Most folks who know will be the first to tell you If there is something they don't know.
Anyway, it appears you got hold of somebody's work who did not know. Some of their work gets by and some causes unbelievable problems.
I believe you are trying to do the right thing, but it looks like you may need to correct some of the other guy's work.
The ground cable is usually connected to some real metal of the engine like the engine block or starter mounting. Who knows that may be why a push button start was jury rigged. Grounds are important and we at this time don't know if only the engine has a good ground , but not some other electrical parts that may affect ignition. Let us know what happens after you replace those parts you have in mind, thanks.

This reminds me of a 49 chevy truck I once had . The previous owner did some wiring under the dash and didn't pay attention to color codes just used what ever color he had .What a mess .A green wire would change to red after the electrical tape splice .So the red wire was really suppose to be green on the fire wall ,LOL. There were more than that .Well I was thinking about taking it all out and starting with a new wire harness but I ended up selling the truck .