Revised answer
To other readers, another issue was the OP used the wrong route.
So make sure to make the API request to the correct route, which is a combination of the first two parameters for register_rest_route() (namespace + route base). So with register_rest_route( \'uheme/v1\', \'/menu\' ), the route is uheme/v1/menu. Additionally:
With permalinks enabled, you can get a pretty endpoint URL (for the above route) like https://example.com/wp-json/uheme/v1/menu
But regardless permalinks enabled or disabled, the endpoint is also accessible at index.php?rest_route=<route> as in https://example.com/index.php?rest_route=/uheme/v1/menu
And to OP, if a callable/callback is in the form of array( __CLASS__, \'method_name\' ) (or maybe \'My_Class::method_name\'), then you need to define the class method as static (e.g. public static function method_name()) to prevent PHP errors. :)
And note that you also made a mistake with that \'method\' => \'GET\' which should be \'methods\' => \'GET\' — note the plural "methods". For GET (request) method, that method would work because GET is the default method, but if you were only allowing POST method and you used \'method\' => \'POST\', then that wouldn\'t work — POST would never be allowed! So once again, the correct array key is methods.
Original answer
Your original class (without extending WP_REST_Routes) worked for me, after I changed the $this to __CLASS__:
\'callback\' => array($this, \'_rest_menu\') // I changed this
\'callback\' => array(__CLASS__, \'_rest_menu\') // to this
Without that change, I got this error:
{"code":"rest_invalid_handler","message":"The handler for the route is invalid","data":{"status":500}}
So make sure you provide a valid callable/callback as the route handler. Or that be aware that in your utheme_routes() method, the $this is not available because you didn\'t do something like $instance = new Uheme_Rest_Routes; $instance->init();.
Update: You can also try changing the \'method\' => \'GET\' to \'methods\' => \'GET\'. And be sure the route is /wp-json/uheme/v1/menu.