This is entirely too long. As a new person, you don't really know any of the processes so may feel unequipped to push back. Great news about the Navy though is everything is a process documented somewhere, and google can be your friend.
I googled "Navy NOSC gain process." Technically the reserve centers are now called NRC vice NOSCs, but a lot of the documentation has not caught up yet.
Resperman 1100-050 "Personnel Gain Transactions" comes back. First it says "Navy recruiters are required to submit a gain package for each affiliation to the NRA within 3 working days of the effective date of affiliation." You can double check with your recruiter but considering its 4 months later, I'm relatively sure they've done this.
Then, it give directions for the NRC to do the actual gain. This is basically a data entry process.
Who have you been in touch with at the NRC?
I would call again, explain your situation. I believe you want to talk to the manpower department. If any pushback is provided, "can you confirm if you are in possession of my gain package?" This is not a complicated process for them, its literally just a matter of someone doing it.
If the response is not satisfactory, email the NRC XO. "XO, I am a new accession having commissioned in November. I know I need to attend ODS within 1 year of commissioning, but I am still not gained into the NRC and am not getting a clear answer on what the hold up is. Can you provide me some guidance on how to proceed?"
I can't imagine NRC leadership is comfortable with a new accession not being gained for four months, and will likely try to get it resolved quickly.