Place the preventDefault as the first line, then if you want the form to submit, call the submit method on the form element. By calling the form element's submit method and not the one defined by jQuery, it will bypass the jQuery bound submit event handler.
$('#formID').submit(function (event) {
event.preventDefault();
var form = this;
var InspectionDate = $('#datepicker').val().split('/');
InspectionDate.reverse();
InspectionDate = InspectionDate.join('-');
InspectionHour = $('#time_hour').val();
InspectionMinutes = $('#time_minutes').val();
var InspectionDateTime = InspectionDate + ' ' + InspectionHour + ':' + InspectionMinutes + ':00';
$.ajax({
type: "POST",
url: "ajax_booking_check.php",
data: 'InspectionDateTime=' + InspectionDateTime,
cache: false,
success: function (response) {
if (response = 1) {
alert("An appointment for the selected Date and Time already exists.\n\nDouble Bookings are not possible.\n\nPlease refer to the calender to find an available appointment.");
} else {
form.submit();
}
}
});
});