To get this value, use the wp.data module. 

    const template = wp.data.select( 'core/editor' ).getEditedPostAttribute( 'template' );

Since this can be changed in the document settings, you would likely need to "subscribe" and run a callback whenever this changes. 

For example:

    const { select, subscribe } = wp.data;
    
    class PageTemplateSwitcher {
    
      constructor() {
        this.template = null;
      }
    
      init() {
    
        subscribe( () => {
    
          const newTemplate = select( 'core/editor' ).getEditedPostAttribute( 'template' );
    
          if ( newTemplate && newTemplate !== this.template ) {
            this.template = newTemplate;
            this.changeTemplate();
          }
    
        });
      }
    
      changeTemplate() {
        // do your stuff here
        console.log(`template changed to ${this.template}`);
      }
    }
    
    new PageTemplateSwitcher().init();