My Grandfather was started out in the Somerset Light Infantry and ended the war in the London Scottish Regiment. I recalled a photo of him (I dont have it now) that showed a black cat badge on his shoulder. After doing some research recently I found the to be the 56th London Regiment, its Dick Whittingtons cat.
I have found that the London Scottish regiment was actually part of this, so the two do tie together. I do still have his full uniform, Kilt, sporran and all!