Thornycroft small vessel construction for the Navy seems to have been concentrated on the Thames at the Hampton Launch Works. Production of the fast and large vessels being at Southampton. Have you tried The Medusa Trust, they may have some information.
You gave the overall length as 40 feet, If you have the measurement for the beam and moulded depth It may be possible to pinpoint the type and builder. I think at 40 feet that rules out a ships workboat.
You probably will have to go bilge diving to find numbers on the keel. There will be some information deep down. Major bulkheads may reveal some information. Boat/shipbuilders, always leave there footprint. Its partly pride in their work. I think early pinnace power after steam was Perkins S6 and Thornycroft db2.
It may just be me but just the hull construction does not look Thornycroft.. There were traditional boat builders building work boats for both harbour transport and warships, when these warships were disposed of the workboats were sometimes used on harbour duties. It will have an interesting history.
In the Google shot it should be in plain view slightly left of the tail of TK 620. TK 620 seems to have been in service until 1957 when it is listed as SS 22-2-57.
Google is one aid, sometimes it is not about the answer it is about the journey. Other information and reminiscences are sometimes priceless, its about being involved.