In the same way that soldiers who had served their time with the colours could go on to serve with the Army Reserve, there was similarly the Royal Fleet Reserve, comprising not only the naval equivalent, but also the RNR and RNVR elements too.
Whilst it was possible for some soldiers to be Time Expired, and thus discharged from the British Army up to 1916, it was not the same for sailors, owing to the King’s proclamation of 3 August 1914:
https://www.greatwarforum.org/topic/233998-when-were-naval-reservists-called-up/?tab=comments#comment-2438186
At the time of mobilisation in August 1914, a number of men were posted to the RN Depot (Reserves) at Crystal Palace. Those personnel had index cards to capture their basic details, which have survived and are in the ADM 339 set of records. About 48,000 of these records relate to the RNVR, 1,000 relate to the RNR, 200 or so relate to soldiers (armourer sergeants) and medical staff. The remaining 1,000 or so records relate to men of the RFR. A lot of these men would serve in the Royal Naval Division.
This RFR group can be broken down into three subsets:
Old sailors
Younger sailors
Former short service
(With regard to the third category, those that signed up for “Special Service” did not spend the full 12 years at sea, but a shorter period of 5 years and the remaining seven years were in the Royal Fleet Reserve. Their service numbers have a “SS” prefix.)
In addition to the card index records in ADM 339, their actual service records are in ADM 188. These can be accessed individually via The National Archives website. Alternatively, FindMyPast have a subset for sailors who enlisted after 1900, and Ancestry have the entire dataset.
The late Paul Benyon’s excellent website has an interesting page about the Royal Fleet Reserve in January 1914
https://web.archive.org/web/20190810114559/http://www.pbenyon.plus.com/RFR/RFR.html