My alternator went out so I replaced it and now it won’t start I wired it the exact same way it was wired before
It doesn’t sound like the engine has any compression.
Tester
Was it running fine before the alternator failure?
Granted my hearing is nowhere near what it used to be but I agree with Tester that it sounds like there is no compression and the engine is freewheeling.
I can’t think of a way that replacing the alternator would result in total loss of compression. If the belt path got mixed up it might cause the engine to not start, but correcting the belt path should fix that problem. So double check the belts are all in the same orientation as before, and everything that is supposed to be rotating during cranking actually is.
My guess is you aren’t getting spark, possibly due to some miss-wiring problem, mixing up connectors, or leaving a connector disconnected, etc. So first step is to use a spare spark plug to check for spark during cranking. If there’s no spark one thing is certain: the engine won’t start.
Less likely – assuming your spark test is ok – something is causing the cylinders to flood with too much gasoline. This can result in an engine that sounds like it has reduced compression. Sometimes when doing repeated testing with the ignition system disabled, that can inadvertently flood the cylinders. To check for that, remove a spark plug and see/smell if the tip is wet w/fuel.
How so?
I’m not questioning your knowledge and expertise . . .
I’d just like you to clarify your position 
The starter drive isn’t engaging because the small wire to the starter solenoid is missing, there is a blue connector with no wire in it.
It appears that a remote starter solenoid has been installed but the solenoid on the starter must be functioning for the starter drive to engage.
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The incorrectly positioned belt is pinching or chaffing a wire for example.