If you are trying to display a list of post that belongs to two or more taxonomy terms, you are using the wrong URL. If your are usgin pretty permalinks, it should be:
www.example.com/service-provider/?service=electricians&area=north-shore-lower
If you are not using pretty permalinks, it should be:
www.example.com/?post-type=service-provider&service=electricians&area=north-shore-lower
Anyway, you are expected to be in multitax archive page and you can’t check it with is_tax(), a function intended to check for single taxonomy archive. You can use some alternatives, like checking the actual $wp_query object:
add_filter('template_include', 'wpse_multietax_template');
function wpse_multietax_template( $template ){
global $wp_query;
if( (isset($wp_query->query_vars['service']) && isset($wp_query->query_vars['area'])) || isset($wp_query->query_vars['service']) ) {
$template = locate_template( array( 'taxonomy_service.php' ) );
} else if ( isset($wp_query->query_vars['area']) && !isset($wp_query->query_vars['service']) ) {
$template = locate_template( array( 'taxonomy_area.php' ) );
}
return $template;
}