You can return a promise inside checkOtherBookings. AngularJS parser automatically deals with promises. So your code would look like this:

$scope.checkOtherBookings = function(bookingType, date) {
    var deferred = $q.defer();
    $scope.$watch('bookings', function(bookings) {
        if (!bookings) return;
        var newBookingArray = [];
        for(var i = 0; i < $scope.bookings.length; i++) {
            if($scope.bookings[i].Booking.Type == bookingType) {
                var tmpDateFrom = $scope.bookings[i].Booking.DateFrom;
                var tmpDateTo = $scope.bookings[i].Booking.DateTo;
                if(date >= tmpDateFrom && date <= tmpDateTo) {
                    newBookingArray.push($scope.bookings[i]);
                }
            }
        }
    
        deferred.resolve(newBookingArray);
    });
    return deferred.promise;
};

I've created a plunker that demonstrates the technique here: demo link.

You can return a promise inside checkOtherBookings. AngularJS parser automatically deals with promises. So your code would look like this:

$scope.checkOtherBookings = function(bookingType, date) {
    var deferred = $q.defer();
    $scope.$watch('bookings', function(bookings) {
        if (!bookings) return;
        var newBookingArray = [];
        for(var i = 0; i < $scope.bookings.length; i++) {
            if($scope.bookings[i].Booking.Type == bookingType) {
                var tmpDateFrom = $scope.bookings[i].Booking.DateFrom;
                var tmpDateTo = $scope.bookings[i].Booking.DateTo;
                if(date >= tmpDateFrom && date <= tmpDateTo) {
                    newBookingArray.push($scope.bookings[i]);
                }
            }
        }
    
        deferred.resolve(newBookingArray);
    });
    return deferred.promise;
};

I've created a plunker that demonstrates the technique here: demo link.

Updated:

The above approach works for AngularJS 1.0.x. The parser of AngularJS 1.2RC (and possibly 1.1.x) handles function returning promise differently, specifically it doesn't return the promise but immediately returning the internal $$v of the promise, which is undefined because it's not yet resolved. If you use 1.2, I suggest get rid of ng-init and try one of the approaches below.

Approach #1:

$scope.$watch('bookings', function(bookings) {
    if (!bookings) return;
    // filter bookings, then set $scope.otherBookings = filteredList
});

Approach #2:

$scope.getOtherBookings = function() {
   // return filter list here
}

<span ng-repeat="otherBooking in getOtherBookings()">

Approach #3:

<span ng-repeat="otherBooking in bookings | customFilterFunction">