I know more about Army than Navy but bare with me.
If he does not want a Queens Commission due to exams, then yes the other option is to start as a private or equivelant in the Navy.
His progression through ranks will depend on his behaviour/attitude/work ethic and his all round performance. Now some people take years to progress through ranks, some never do progress. You could leave after 22 years service where you started, as a sergeant or top of the non commissioned ranks. Some do cross over to a Queen commissioned rank, if they are seen to have "officer potential" and are put forward by their senior officers to do so.
Nobody can really say if your child will enjoy the Navy, it's all really a personal experience as you can imagine. I have friends who served 3 years hated it and others who are still going after 20 and loving it.
Think carefully, if your DC has gotten to the point of sitting the exams for entry to the Royal Naval College, then he has shown officer potential in his previous interviews. Now a Officers life is very different to that of a soldiers, if anything slightly better. Better pay better lifestyle in general including better living quarters and so forth. It's not fair no but it's just the way it is.
A life in the Armed Forces can turn in to a wonderful career path, he must think carefully about where and what he wants from his career, and where to begin it. Does he have a flare for people management? Is he a good communicator? Is he confident in himself and the decisions he makes? All I would say is think carefully, if he truly believes a career as a officer is not for him, then by all means go for what will suit him and what he believes he will enjoy.
At the end of the day, you must be happy with your work, and when it comes to the armed forces your job is your life. You live at base/on board, everything can sometimes feel micro managed by the job. But it is a highly rewarding career with many opportunities regardless of where you start on the ladder.
Hope I have helped a little x