JavaScript/promise, multiple async calls to sharepoint

You are using the deferred antipattern. Every asynchronous action should get its own promise that represents the result, only then you can properly compose them. Currently you have something like an array of lists of objects, but it needs to become an array of promises to work with Q.allSettled.

Actually, you don't even need Q.allSettled, as you don't fire multiple queries at the same time but only a single one.

So lets build a helper function for the promise creation:

 function load(context, objs, args) {
     if (!Array.isArray(objs))
         context.load(objs, args);
     else
         for (var i=0; i<objs.length; i++)
             context.load(objs[i], args[i]);
     var deferred = $q.defer();
     context.executeQueryAsync(function(sender, args) {
         deferred.resolve(objs); // sender, args don't seem to be helpful
     }, function(sender, args) {
         deferred.reject(args);
     });
     return deferred.promise;
}

Now you can use it like this:

function getWebs($q) {
    //App context etc..
    var web = appContextSite.get_web();
    var subWebs = web.getSubwebsForCurrentUser(null);

    return load(context, subWebs, 'Include(Url, Created, Title, Lists)')
//  ^^^^^^
    .then(function(loadedSubWebs) {
        var enumSubWebs = loadedSubWebs.getEnumerator();
        var list = [],
            loadList = [];

        while (enumSubWebs.moveNext()) {
            var subWeb = enumSubWebs.get_current(),
                subWebUrl = subWeb.get_url();

            var obj = {
                listItem: subWeb.get_lists().getByTitle('Custom List').getItemById(1),
                webTitle: subWeb.get_title()
            };
            list.push(obj);
            loadList.push(obj.listItem);
        }
        return load(context, loadList).then(function(loadedList) {
//      ^^^^^^
            for (var i = 0; i < loadedList.length; i++) {
                console.log(loadedList[i].get_fieldValue().Title);
            }
            return list;
        });
    }, function(err) {
        // the load failed. `err` will be the `args` passed to `reject`
        console.log("list doesn't exist in this web");
        // you can still resolve the promise:
        return [];
        // or alternatively rethrow the exception:
        throw err;
    });
}

Here's something very similar to Bergi's answer but independently derived.

The main difference is that I have left the context.load() statements behind in the main routine and promisified just context.executeQueryAsync(). This is less rational than Bergi's load() but makes for a simpler helper and simpler main routine.

var getW = getWebs($q).then(function (results) {
    console.log(results); // log the `list` derived from the first execute.         
});

function executeQueryPromisified(context) {
    /* Ref 1 */
    var deferred = $q.defer();
    context.executeQueryAsync(
        function(sender, args) {
            deferred.resolve(args);
        },
        function(sender, args) {
            deferred.reject(args);
        }
    );
    return deferred.promise;
};

function getWebs($q) {
    var web = appContextSite.get_web(),
        subWebs = web.getSubwebsForCurrentUser(null);
    context.load(subWebs, 'Include(Url, Created, Title, Lists)');
    return executeQueryPromisified(context).then(function(args) {
        var enumSubWebs = subWebs.getEnumerator(),
            list = [], subWeb, obj;
        while (enumSubWebs.moveNext()) {
            subWeb = enumSubWebs.get_current();
            obj = {
                'listItem': subWeb.get_lists().getByTitle('Custom List').getItemById(1),
                'webTitle': subWeb.get_title()
            };
            list.push(obj);
            context.load(obj.listItem);
        }
        return executeQueryPromisified(context).then(function() {
            for (var i = 0; i < list.length; i++) {
                console.log(list[i].listItem.get_fieldValue()['Title']);
            }
            return list;
        });
    }).catch(function (args) {
        /* Ref 1 */
        console.error('Request failed. ' + args.get_message() + '\n' + args.get_stackTrace());
    });
};

Ref 1: Sharepoint documentation; SP.ClientContext.executeQueryAsync method