The this pointer is different inside the callback function. So, you need to save it into a different variable. Then, use that variable instead of this inside your callback. For a Lightning helper, I save it to one called helper

save : function(component) { 
// ... 
 var fr = new FileReader();
 var helper = this;
 fr.onload = function() {
        var fileContents = fr.result;
        var base64Mark = 'base64,';
        var dataStart = fileContents.indexOf(base64Mark) + base64Mark.length;
        fileContents = fileContents.substring(dataStart);
        alert("WORKING"); //check at this point- alert works
        helper.upload(component, file, fileContents);
        alert("DID IT WORK?"); // This alert not shown so upload call not working
    };
    fr.readAsDataURL(file); //fr.onload function triggered from here
},

upload: function(component, file, fileContents)
{ alert("Here");  //NEVER CALLED              
} 

See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Operators/this

The this pointer is different inside the callback function. So, you need to save it into a different variable. Then, use that variable instead of this inside your callback. For a Lightning helper, I save it to one called helper

save : function(component) { 
    var maxFileSize = 750000;
    var fileInput = component.find("file").getElement(); 
    var file = fileInput.files[0];  
            if (file.size > maxFileSize) {
        alert('File size cannot exceed ' + maxFileSize + ' bytes.\n' +
              'Selected file size: ' + file.size);
        return;
    }

 var fr = new FileReader();
 var helper = this;
 fr.onload = function() {
        var fileContents = fr.result;
        var base64Mark = 'base64,';
        var dataStart = fileContents.indexOf(base64Mark) + base64Mark.length;
        fileContents = fileContents.substring(dataStart);
        alert("WORKING"); //check at this point- alert works
        helper.upload(component, file, fileContents);
        alert("DID IT WORK?"); // This alert not shown so upload call not working
    };
    fr.readAsDataURL(file); //fr.onload function triggered from here
},

upload: function(component, file, fileContents)
{ alert("Here");  //NEVER CALLED              
} 

See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Operators/this

The this pointer is different inside the callback function. So, you need to change it like this:

save : function(component) { 
// ... 
 var fr = new FileReader();
 var helper = this;
 fr.onload = function() {
        var fileContents = fr.result;
        var base64Mark = 'base64,';
        var dataStart = fileContents.indexOf(base64Mark) + base64Mark.length;
        fileContents = fileContents.substring(dataStart);
        alert("WORKING"); //check at this point- alert works
        helper.upload(component, file, fileContents);
        alert("DID IT WORK?"); // This alert not shown so upload call not working
    };
    fr.readAsDataURL(file); //fr.onload function triggered from here
},

upload: function(component, file, fileContents)
{ alert("Here");  //NEVER CALLED              
} 

See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Operators/this

The this pointer is different inside the callback function. So, you need to save it into a different variable. Then, use that variable instead of this inside your callback. For a Lightning helper, I save it to one called helper

save : function(component) { 
// ... 
 var fr = new FileReader();
 var helper = this;
 fr.onload = function() {
        var fileContents = fr.result;
        var base64Mark = 'base64,';
        var dataStart = fileContents.indexOf(base64Mark) + base64Mark.length;
        fileContents = fileContents.substring(dataStart);
        alert("WORKING"); //check at this point- alert works
        helper.upload(component, file, fileContents);
        alert("DID IT WORK?"); // This alert not shown so upload call not working
    };
    fr.readAsDataURL(file); //fr.onload function triggered from here
},

upload: function(component, file, fileContents)
{ alert("Here");  //NEVER CALLED              
} 

See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Operators/this