Home Fleet Grey was a very dark shade of grey which by the time the TIDs were built in 1943 was no longer used. There was a standard grey scheme used for merchant ships which I would assume tugs would be described as because they weren't RN vessels but MOWT.
MSS > Merchant Ships Side & MSD > Merchant Ships Deck and later Light MSS. If you look at David Williams's book on camouflage, it says that AP507A and AP507C wre used initially, A being Home Fleet Grey & C being Med Fleet Grey. I have the book on Empire Tugs and many of them are shown mid and late war as being in a light grey colour, so may be the TIDs were this light grey. Sorry I don't have any books on the wartime TIDs. A photo of Pasteur in William's book shows her as light gery with white mast tops. So may be AP507C could be the TID's color.