a.js

  constructor() {
    this.Header = new Header();
    this.Log = new Log();
    console.log(typeof this.messageHandler); // function
    this.UserInput = new UserInput(this.messageHandler);
  }

b.js

  constructor(messageHandler) {
    console.log(typeof messageHandler); // function
    this.newMessageHandler = messageHandler;
    console.log(typeof this.newMessageHandler); // undefined
  }

  keyupHandler(e) {
    if(e.keyCode == 13) { // Enter key
      console.log(this.newMessageHandler); // undefined
      this.newMessageHandler({
        "user": "a",
        "room": "b",
        "type": "message",
        "body": e.target.value,
        "date_added": "1"
      });