The National Archives' research guide Looking for records of a person in the Royal Naval Volunteer Reserve warns:
Many records are online, but it is important to note that not all
So this could be a coverage problem (your Dad is right but your grandfather's records didn't survive), or it could be that whatever material survives is not yet online.
Ancestry.co.uk's collection UK, Navy Lists, 1888-1970 doesn't have any entries for 1940 in the browse list (see the right-hand side of the page).
The World War Naval Ships forum has a Member article on Royal Naval Reserve (Personnel) with an overview of record collections that can be searched (slightly out of date; note Documents Online at the National Archives has closed since the article was written).
I see two possible avenues of research:
Find other sites which mention the Cutty Sark
In 1940 she was converted into a submarine tender and attached to the
3rd Submarine Flotilla. She was manned by the Royal Naval Patrol Service.
She was paid off on 23 August 1944.
Unfortunately there is no footnote to show which of the references this information might have come from.
My suggestion would be to continue your research of the Cutty Sark herself, and of other ships that were part of the 3rd Submarine Flotilla, and make careful note of what sources are used by the authors of the books, websites, and other material you find.
Find other people who served on the Cutty Sark
In the references for the Wikipedia article on the Royal Naval Patrol service, there is a link to the Royal Naval Patrol Service Association.
Looking for more information about the Association, I found a writeup on the site Culture 24 which includes this caution:
The RNPS Association holds no records about individual members'
service or medal entitlements. The main record of service is the form
S-459 which records name, next of kin, postings, rating, etc. and is a
potted account of each member's entire service. The original document
was handed over to the member on discharge and there is no duplicate
on file. Only brief details are likely to be recorded in the Ministry
of Defence archives and so this prime record is the main source of any
detail and only likely to be found amongst the member's personal
The S-459 may be the form which you already have, from which you got his service number. If that's the case, you may not be able to find much more if you are searching for him by name.
However, if you have the names of your Grand-Dad's fellow officers on the Cutty Sark, you can use a search for their names as a test case. If more collections of records come online, you will have more than one name to try, to see what the coverage of that particular collection might be. You might also find memoirs that mention your grand-dad that weren't indexed with his name (or for which the index was faulty). This technique has been especially helpful when searching historical newspapers, books, and other documents that might have been indexed with OCR (Optical Character Recognition).
Studying people in groups is often called "cluster' genealogy or the FAN principle (FAN = friends, associates, neighbors). For an overview of the FAN principle, see QuickSheet: The Historical Biographer’s Guide to Cluster Research (the FAN Principle) by Elizabeth Shown Mills.
Widening your search for other references to the ship, and to the group of people who might have served with him, will help fill in the missing bits of the picture.