For our Android and iOS clients, this is not as easily possible yet as we would like. tus-js-client has good support for retrieving responses, but our other clients not yet. For the Android client, you could look into Response server as JSON · Issue #89 · tus/tus-java-client · GitHub. For the iOS client, I will forward this question to our iOS developer.
In TUSKit versions 3.1.6 and earlier this is unfortunately not possible. I have just published TUSKit 3.1.7 which will allow developers to catch and inspect the status code for requests that received a non 200 OK HTTP status code.