Why is the form not updating when I select a new sector from the list?

It looks like only your locations drop-down is triggering the ajax query. I believe the issue is that the javascript is listing for a change event on these elements: '#search_keywords, #search_location, .job_types :input, #search_categories, .job-manager-filter' in the code below (line 178 in http://s195004125.websitehome.co.uk/wp-content/plugins/wp-job-manager/assets/js/ajax-filters.js?ver=1.23.13)

$( '#search_keywords, #search_location, .job_types :input, #search_categories, .job-manager-filter' ).change( function() {
    var target   = $( this ).closest( 'div.job_listings' );
    target.triggerHandler( 'update_results', [ 1, false ] );
    job_manager_store_state( target, 1 );
} )

But, the actual id of the element is sector, not search_categories – perhaps try adding ‘#sector’ to the elements listed, so that the javascript is listening for its change event too.