Hello Max, I may be able to shed some light on this problem. The logs for smaller ships for that period have not survived except for a few scattered examples. Spey was not one of the chosen examples. The background story is a sorry tale: in the 1950's someone made a decision way above their pay grade and ordered the destruction of thousands of documents, mostly the logs of smaller ships, because the space was needed. The logs of larger ships went to another location because they were the only ones, in his opinion, that were of potential interest.
The National Archives at Kew is where the surviving logs are kept and you are right, there are no logs from Spey held there. However, depending on what you are looking for, there are other files that may be very useful. The work done by escort and support craft was summarised in a number of places and Spey is recorded as escort in a number of convoys. This is visible in the descriptions that appear in the online catalogue (apologies if you have already established this):
I have narrowed the search to the ADM series of records between 1939 and 1945 using 'SPEY' as 'FIND ALL'. Most of the hits are around the role of SPEY as a convoy escort. Where there are 'Honours and Awards', such as ...
... you will find a report of the action concerned (in this case the sinking of U-386 and U-406) by the commander of the SPEY and possibly other participants. The 'Report of Proceedings' in this one ...
... describes a convoy and includes the part played by SPEY.
There are other sources for more specific issues such as crew, where based, movements etc. that can be viewed in person at Kew. To find these is a little more work because the name SPEY will not appear in their digitised description: the files include SPEY as part of their narrative but are not about SPEY in particular.
So, depending on what you want to know, it is possible the information within the files at Kew may be sufficient. If you want to follow this path of breadcrumbs, send me a PM. I was a researcher at Kew, now retired, and may be able to help you 'cut to the chase'.
Hope this helps, and again, sorry if I am covering ground already familiar to you.