Example with both attempts

components.async-image-wrapper.js

import Ember from 'ember';

export default Ember.Component.extend({
  imageLoaded: 'image loading',
  imageWidth: 'width loading',
  
  // purely to make sure image loads each time and
  // is not cached for testing
  volatileSrc: Ember.computed('src', function(){
    return this.get('src') + '?cache=' + new Date().toISOString();
  }),
  
  actions: {
    imageLoaded(message, width) {
      if (!this.isDestroyed) {
        this.set('imageLoaded', message);
        this.set('imageWidth', width);
      }
    }
  }
});

components.async-image.js

import Ember from 'ember';

function loadImagePromise(src) { 
  return new Ember.RSVP.Promise( function(resolve, reject) {
    var img = new Image();
    
    // this waits, but the img width is wrong
    img.onload = Ember.run(resolve(img));
    
    // img and event.target now return the
    // correct width (412) of the image
    // but doesn't wait for the resolve anymore
    
    // img.onload = (event) => {
    //   Ember.run(() => {
    //     resolve(img);
    //   });
    // }
    
    img.src = src;
  });
}

export default Ember.Component.extend({
  tagName: 'img',
  attributeBindings: ['src'],
  
  loadImage() {
    let src= this.get('src');
    
    loadImagePromise(src)
    .then(img => {
      this.get('onImageLoad')("image loaded", img.width);
    })
    .catch(error => {
      Ember.Logger.log('error', error);
      this.get('onImageLoad')("image could not load");
    });
  },

  didInsertElement() {
    Ember.run.scheduleOnce('afterRender', this, this.loadImage);
  }
});

controllers.application.js

import Ember from 'ember';

export default Ember.Controller.extend({
});

styles.app.css

body {
  margin: 12px 16px;
  font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif;
  font-size: 12pt;
}

h1, h2 {
  //display: none;
}

templates.application.hbs

<h1>Testing async behaviour</h1>
<h2>Using wait() in integration tests</h2>
<p>I thought that by using a promise or the run loop then the wait helper would pause before checking the assertion, but it only works with run.later and I can't work out why.</p>
{{async-image-wrapper src='http://emberjs.com/images/tomsters/sandiego-zoey.png'}}

templates.components.async-image-wrapper.hbs

{{async-image src=volatileSrc onImageLoad=(action 'imageLoaded')}}
<p id="image-loaded-text">{{imageLoaded}}</p>
<p>width: <span id="image-width-text">{{imageWidth}}</span></p>

tests.helpers.resolver.js

import Resolver from '../../resolver';
import config from '../../config/environment';

const resolver = Resolver.create();

resolver.namespace = {
  modulePrefix: config.modulePrefix,
  podModulePrefix: config.podModulePrefix
};

export default resolver;

tests.integration.components.async-image-wrapper-test.js

import { moduleForComponent, test } from 'ember-qunit';
import wait from 'ember-test-helpers/wait';
import hbs from 'htmlbars-inline-precompile';

moduleForComponent('async-image-wrapper', 'Uses Promise', {
  integration: true
});

test('If image found update message to success', function(assert) {
  assert.expect(2);
  
  this.render(hbs`{{async-image-wrapper src='http://emberjs.com/images/tomsters/sandiego-zoey.png'}}`);
  
  return wait().then( () => {
    assert.equal(this.$('#image-loaded-text').text(), 'image loaded');
    assert.equal(this.$('#image-width-text').text(), '412');
  });
});

tests.test-helper.js

import resolver from './helpers/resolver';
import {
  setResolver
} from 'ember-qunit';

setResolver(resolver);

twiddle.json

{
  "version": "0.10.4",
  "EmberENV": {
    "FEATURES": {}
  },
  "options": {
    "use_pods": false,
    "enable-testing": true
  },
  "dependencies": {
    "jquery": "https://cdnjs.cloudflare.com/ajax/libs/jquery/1.11.3/jquery.js",
    "ember": "2.6.0",
    "ember-data": "2.6.0",
    "ember-template-compiler": "2.6.0"
  },
  "addons": {}
}