Você só precisa agrupar suas pontuações por usuário:
SELECT @p:[email protected]+1 AS position, t.*
FROM (
SELECT user.user_id,
user.user_name,
IFNULL(SUM(score.score_points),0) AS total_points
FROM user LEFT JOIN score ON user.user_id = score.score_user_id
GROUP BY user.user_id
ORDER BY total_points DESC
) AS t JOIN (SELECT @p:=0) AS initialisation
Veja em sqlfiddle .
