Unless they joined another existing party I'd assume they would form a new political group ("Independent Labour" or something more pertinent, e.g. "anti-genocidal Labour"). Then unless the remaining Labour group wants to enter a formal coalition with someone else, you might expect that most business just goes on as before.
In Oxford, the new independents would have to side with one of the other parties to change much. Labour only needs two non-Labour votes to have a majority (from former Labour, LibDem, or Green). There are no Tory councillors.
Tories could only take over in a council if former Labour councillors voted with the Tories.