What was my father’s service record number? His details: A/Surgeon Lt-Cdr Eric Edwin Kingsley Kilvert, RNR. He joined up in Feb 1940 at Chatham (HMS Pembroke)
Ancestry has "UK, Navy Lists, 1888-1970" - these include "multiple volumes of the Navy List, the Royal Navy’s official published list of officers." There are 4 records for your father (1941, 42, 44 and 1945 - I'd guess he's in the 1943 as well but it's not indexed?)
p.1639 of Volume II of the July 1945 list (I don't know how often the paper versions were published) has him under the Temporary Surgeon Lieutenants of the Royal Naval Volunteer Reserve (note - RNVR not RNR - the two are separate at this time). He's described as
Eric Edwin Kingsley Kilvert, MB, ChB, 2 Feb 1940 (the date will be his seniority in that rank)
There is also a note to say that he is a "Temporary Acting Surgeon Lieutenant Commander" - no, I don't know what the subtleties of being a Temporary Acting anything are, when compared to an Acting anything or a Temporary anything! No date for that.
And that's it. For other details on how to research RNVR personnel see http://www.nationalarchives.gov.uk/records/looking-for-person/royalnavalvolunteerreserve.htm
If you intend to apply for his service record (as next-of-kin, or with permission of NoK) then the link to the MoD is on the TNA page. Although the forms have a section for the official service number, it is my suspicion that quoting that stuff from the Navy List - in full - will suffice to find him. I can't be certain of this but I can say that although Army Officers in WW1 didn't have any numbers, the War Department still managed to administer them, so there were ways and means then.
(Delicious irony - the editions of the Royal Navy List on Ancestry that I refer to, are from an academic institution in Germany!)