I just tested changing the time of the appointment too, and this is not reflected in the Google Calendar either...

After debugging I've found the call Runner::run throws an exception:

Error calling PUT https://www.googleapis.com/calendar/v3/calendars/simon%40axcultures.com/events/c4gaod83c0r82efkbb7nuo48hg?sendNotifications=false: (404) Not Found

Call stack:

Runner.php:176, AmeliaGoogle_Task_Runner->run()
REST.php:46, AmeliaGoogle_Http_REST::execute()
Client.php:593, AmeliaGoogle_Client->execute()
Resource.php:240, AmeliaGoogle_Service_Calendar_Events_Resource->call()
Calendar.php:1742, AmeliaGoogle_Service_Calendar_Events_Resource->update()
GoogleCalendarService.php:393, AmeliaBooking\Infrastructure\Services\Google\GoogleCalendarService->updateEvent()
GoogleCalendarService.php:226, AmeliaBooking\Infrastructure\Services\Google\GoogleCalendarService->handleEvent()
AppointmentEditedEventHandler.php:137, AmeliaBooking\Infrastructure\WP\EventListeners\Booking\Appointment\AppointmentEditedEventHandler::handle()
AppointmentEventsListener.php:69, AmeliaBooking\Infrastructure\WP\EventListeners\Booking\Appointment\AppointmentEventsListener->handle()
Emitter.php:217, call_user_func_array:{D:\www\cac\wp-content\plugins\ameliabooking\vendor\league\event\src\Emitter.php:217}()
Emitter.php:217, League\Event\Emitter->invokeListeners()
Emitter.php:169, League\Event\Emitter->emit()
DomainEventBus.php:40, AmeliaBooking\Domain\Events\DomainEventBus->emit()
UpdateAppointmentController.php:68, AmeliaBooking\Application\Controller\Booking\Appointment\UpdateAppointmentController->emitSuccessEvent()
Controller.php:125, AmeliaBooking\Application\Controller\Booking\Appointment\UpdateAppointmentController->__invoke()
RequestResponse.php:40, call_user_func:{D:\www\cac\wp-content\plugins\ameliabooking\vendor\slim\slim\Slim\Handlers\Strategies\RequestResponse.php:40}()
RequestResponse.php:40, Slim\Handlers\Strategies\RequestResponse->__invoke()
Route.php:281, Slim\Route->__invoke()
MiddlewareAwareTrait.php:117, Slim\Route->callMiddlewareStack()
Route.php:268, Slim\Route->run()
App.php:503, Slim\App->__invoke()
MiddlewareAwareTrait.php:117, Slim\App->callMiddlewareStack()
App.php:392, Slim\App->process()
App.php:297, Slim\App->run()
ameliabooking.php:167, AmeliaBooking\Plugin::wpAmeliaApiCall()
class-wp-hook.php:287, WP_Hook->apply_filters()
class-wp-hook.php:311, WP_Hook->do_action()
plugin.php:484, do_action()
admin-ajax.php:184, {main}()

Furthermore, I was successful in changing the time if I kept the original employee - but once I changed the employee no other changes were synced...

Updated the code myself in Google CalendarService.php and it seems to be working pretty well:

public function handleEvent($appointment, $commandSlug, $originalEmployeeId = null)
    {
        /** @var ProviderRepository $providerRepository */
        $providerRepository = $this->container->get('domain.users.providers.repository');
        $appointmentStatus = $appointment->getStatus()->getValue();
        $provider = $providerRepository->getById($appointment->getProviderId()->getValue());
        if ($provider->getGoogleCalendar() && $provider->getGoogleCalendar()->getCalendarId()->getValue()) {
            $this->authorizeProvider($provider);
            switch ($commandSlug) {
                case AppointmentAddedEventHandler::APPOINTMENT_ADDED:
                case BookingAddedEventHandler::BOOKING_ADDED:
                    // Add new appointment or update existing one
                    if (!$appointment->getGoogleCalendarEventId()) {
                        $this->insertEvent($appointment, $provider);
                    } else {
                        $this->updateEvent($appointment, $provider);
                    }
                    // When status is pending we must first insert the event to get event ID
                    // because if we update the status later to 'Approved' we must have ID of the event
                    if ($appointmentStatus === 'pending' && $this->settings['insertPendingAppointments'] === false) {
                        $this->deleteEvent($appointment, $provider);
                    }
                    break;
                case AppointmentEditedEventHandler::APPOINTMENT_EDITED:
                case AppointmentTimeUpdatedEventHandler::TIME_UPDATED:
                case AppointmentStatusUpdatedEventHandler::APPOINTMENT_STATUS_UPDATED:
                case BookingCanceledEventHandler::BOOKING_CANCELED:
                    if ($appointmentStatus === 'canceled' || $appointmentStatus === 'rejected' ||
                        ($appointmentStatus === 'pending' && $this->settings['insertPendingAppointments'] === false)
                    ) {
                        $this->deleteEvent($appointment, $provider);
                        break;
                    }
                    if (($commandSlug == AppointmentEditedEventHandler::APPOINTMENT_EDITED) &&
                      ($originalEmployeeId !== null) &&
                      ($originalEmployeeId != $provider->getId()->getValue())) {
                      $appointment->setProviderId(new \AmeliaBooking\Domain\ValueObjects\Number\Integer\Id($originalEmployeeId));
                      $this->deleteEvent($appointment, $providerRepository->getById($originalEmployeeId));
                      $appointment->setProviderId($provider->getId());
                      $this->insertEvent($appointment, $provider);
                    }
                    else {
                      $this->updateEvent($appointment, $provider);
                    }
                    break;
                case AppointmentDeletedEventHandler::APPOINTMENT_DELETED:
                    $this->deleteEvent($appointment, $provider);
                    break;
            }
        }
    }

You'll need to make sure everyone has edit permissions on eachother's calendars also:

https://stackoverflow.com/questions/14003203/403-forbidden-message-when-calling-the-v3-google-calendar-api-using-a-service-ac

https://stackoverflow.com/questions/11863022/edit-google-calendar-events-from-google-service-account-403/11897793#11897793