How to deal with wife doing this?

Dude, if you can't trust your wife with your child then go.  Rest assured if you were a gal writing almost the same thing, every woman in America would be calling you every dirty word in the book and telling her to get out.

Get a log book and document everything she says and every time she lies or is not there to responsibly pick up her child. Without that log book any complaint you have will just be considered grumbling.

And Garry, if your place of employment tells you you are on night shift 99% of the people on this planet do not have a chance to say no.

best you both work day shift like  normal couples , or maybe she looking for it else where , after all your night shift arent you . your doing so you fix it either way .

It seems she doesn't want to be a mother or a wife.  I'll almost bet a paycheck that she's not far from stepping out on you with another.

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You’ve posted for years about your wife’s abusive behavior, and despite being given all kinds of advice ranging from getting counseling to getting divorced, you’ve done none of it. You also posted a question recently about your job and the issues you’ve had with punctuality and your boss’s irritation with your chronic tardiness, so this is impacting a lot of your life. If you continue to wallow and have no willingness to take active steps to change your situation, you will be unemployed and still married and raising a child in an abusive relationship.

Get help for your codependency issues so that you can finally find the courage to break free. Because yes, you’re a coward to continue to live like this, despite all the signs you’ve had for YEARS that this is an unhealthy and toxic environment not just for you, but also for your son. Don’t make excuses about your finances, or the debts for homes your wife has purchased or cultural differences or any of the other excuses you’ve made for years. Or if you aren’t going to make any changes, accept your lot in life and stop posting pointless questions about it.

If you are both working, can you really not afford the occasional baby sitter? 

If she won't answer her phone, perhaps she is with another man - something you must have considered/. I am tempted to suggest that she may have had some sort of depression - or breakdown even following giving birth. Naturally we do not know anything at all what she is thinking and feeling, but it is a bit of concern that she does absolutely nothing for her family or her home. If asked, would she agree? 

sir, you are headed for complete burnout and disaster (auto crash or work mistake).  You are trying to balance FT position, PT studies, and de facto single parent. You must prioritize rest, work, and baby son, not nursing or wife.  

"then she isn't really worth keeping in my life long term": not if there's a serious communication gap.

"I cannot afford a baby sitter":  that sounds like the most logical and obvious solution.

Please give yourself at least 6 hrs/ 24 of downtime. 

Sounds like she may have mental issues, could be Bipolar?

 Two things to say

Firstly a question.   Why share your marital problem with strangers?

Then some advice.   Seek a marriage enrichment course 

Marriages are worth working on 

maybe you should hire someone to babysit him so you can go to work

maybe you should hire someone to babysit so you can work