Well, I wasn't going to answer - but I was challenged. 
It's quite easy to utilize the built in abilities of promises in order to get this sort of queuing for free. Here is how this conversion will work:
- We convert the callback API to promises on a new object with a promise subclass.
- We add all the promised methods to the subclass itself - so it chains.
- We tell the subclass to execute all the methods in a then, so they'll queue asthenis the queueing mechanism promises have.
 
Note: The promisify and promisifyAll methods I write here - you should grab off NPM - lots of good and fast usages that take a promise constructor.
First, we need a method that converts a callback API to promises:
// F is a promise subclass
function promisify(fn) { // take a function
    return function(...args) {  // return a new one with promises
      return new F((resolve, reject) => { // that returns a promise
         // that calls the original function and resolves the promise
         fn.call(this, ...args, (err, data) => err ? reject(err) : resolve(data));
      });
    };
  } 
Now, let's promisify the whole object:
  function promisifyAll(obj) {
    const o = {};
    for(const prop in obj) {
      if(!(obj[prop].call)) continue; // only functions
      o[prop] = promisify(obj[prop]).bind(obj);
    }
    return o;
  }
So far, nothing new, lots of NPM libraries do this - now for the magic of promises - let's create a method that executes functions on an original object in a then:
function whenReadyAll(obj) {
    const obj2 = {}; // create a new object
    for(const prop in obj) { // for each original object
       obj2[prop] = function(...args) { 
         // return a function that does the same thing in a `then`
         return this.then(() => obj[prop](...args));
       };
    }
    return obj2;
  }
Now, let's wrap things up 
function liquidate(obj) {
  const promised = promisifyAll(obj); // convert the object to a promise API
  class F extends Promise {} // create a promise subclass
  Object.assign(F.prototype, whenReadyAll(promised)); // add the API to it
  return promised; // return it
  // previous code here
}
And that's it, if we want the example to be self contained (again, promise and promisifyAll are provided by a library usually):
function liquidate(obj) {
  const promised = promisifyAll(obj);
  class F extends Promise {}
  Object.assign(F.prototype, whenReadyAll(promised)); // add the API  
  return promised;
  function whenReadyAll(obj) {
    const obj2 = {};
    for(const prop in obj) {
       obj2[prop] = function(...args) { 
         return this.then(() => obj[prop](...args));
       };
    }
    return obj2;
  }
  function promisifyAll(obj) {
    const o = {};
    for(const prop in obj) {
      if(!(obj[prop].call)) continue; // only functions
      o[prop] = promisify(obj[prop]).bind(obj);
    }
    return o;
  }
  function promisify(fn) {
    return function(...args) { 
      return new F((resolve, reject) => {
         fn.call(this, ...args, (err, data) => err ? reject(err) : resolve(data));
      });
    };
  } 
}
Or with a library that does promisify: 
function liquidate(obj) { // 14 LoC
  class F extends Promise {} 
  const promised = promisifyAll(obj, F); // F is the promise impl
  Object.assign(F.prototype, whenReadyAll(promised)); // add the API  
  return promised;
  function whenReadyAll(obj) {
    const obj2 = {};
    for(const prop in obj) {
       obj2[prop] = function(...args) { 
         return this.then(() => obj[prop](...args));
       };
    }
    return obj2;
  }
}
And what's an answer without a demo:
var o = {  // object with a delay callback method
  delay(cb) { 
    console.log("delay"); 
    setTimeout(() => cb(null), 1000); 
  }
};
var o2 = liquidate(o); // let's liquidate it
// and we even get `then` for free, so we can verify this works
var p = o2.delay().then(x => console.log("First Delay!")).
                   delay().
                   then(x => console.log("Second Delay!"));
// logs delay, then First Delay! after a second, 
// then delay and then Second Delay! after a second
Copy paste this to your friendly neighborhood console and see for yourself :)
To prove this preserves state on the original object (it's easy to modify it not to if that's a requirement) let's add an i variable and increment it in delay and see that things work: 
var o = {  // object with a delay callback method
  delay(cb) { 
    console.log("delay", this.i++); 
    setTimeout(() => cb(null), 1000); 
  },
  i: 0
};
var o2 = liquidate(o); // let's liquidate it
// and we even get `then` for free, so we can verify this works
var p = o2.delay().then(x => console.log("First Delay!")).
                   delay().
                   then(x => console.log("Second Delay!", o.i));
//logs:
// delay 0
// First Delay!
// delay 1
// Second Delay! 2