How to add one more query string with search query string

You should use ‘add_query_arg’ function, to do this.

It adds a single key & value or an associative array. You can set a key value to false to remove it from the query. Omitting the old query or the uri (second or third parameter) uses the $_SERVER value.

So in your case doing it like this should work:

... action="<?php echo add_query_arg( 'uid', get_current_user_id(), get_permalink() ); ?>" ...