How does Unqualified Resignation work with regard to timeline and post-approval actions?

I submitted my unqualified and the process was quick. I was discharged within 30 days from filing. But be prepared for extremely low ratings from your supervisors. Ratings in the 20s is not uncommon. I submitted mine in November 1973 and was discharged December 21, 1973, from the Army.

I can't give you the answer for the USMC, but I can tell you how it's done in the Army. You submit your UQR request along with your requested ETS/EAS. Upon approval, that is your separation date. For the Army, it has to be at least 90 days away, that's because of federal laws requiring at least 90 days to separate. There is usually a max amount out you can request, for the Army I think it is one year. It is probably between six and twelve months for the other services.

I did a bit more digging and it seems very much like the Army in process.

Approval is at the department Secretariat level (Secretary of the Navy in your case - "... to the Secretary of the Navy via the CMC (MMSR-5) and the COMMARFORRES" -- I have no clue who those are BTW). You'll likely have to get recommendations from commanders (not just approval though the chain of command) as well.

Resignation vs REFRAD - With the personnel shortages going on in all the services, I think you'll have a huge hill to climb if you have a remaining service obligation (doing a resignation and not serving the rest of your time in the reserve component). REFRAD might not be an option if you still have an ADSO (Active Duty Service Obligation) remaining.