I think I know the photographs you are after, but I haven't come across them yet.
However, in the Pleasures Past video 'Memories of Wirral in the 1940's & 50's' there is a short section around Hamilton Square at 30 minutes in. It shows a bowler hatted gentleman using the Hamilton St Post Office and then pans along the Cleveland St section of the Square to the corner where the Council offices now are, where the old Church, minus its roof, can be seen. It then cuts to the other corner, where the estate agents were, but unfortunately we can't see quite enough to get a glimpse of the shelter. I also think that a longer version of this film appears in another their video's, demonstrating some road safety do's & don'ts, which included a cyclist hanging on to a bus, getting on to a moving bus, and sitting chatting at the lights while they change a few times. I've not come across that one today, but I'll find it.
The shelter was a lean-to canopy with the wall, open at both ends. It had a glass roof with cast iron uprights and the usual tubular railings at the pavement edge. It was long enough to accomodate two busses of the time. I'm sure it survived into the hands of the MPTE.
It is interesting to see the yellow clay bricks used in the wall of the stables. They sometimes appear on buildings in Birkenhead of that time and are occasionally seen as a relief in some Wallasey buildings, but hardly anywhere else around here. There are similar bricks in some other northern towns, but the only other place I can think of that use them in large numbers is in London. Theirs are a slightly different shade, obviously from different clay pits.
The 'modern' white-faced Council building on the corner used to house the Education, Rates and Vehicle Taxation Departments for the Borough, as well as the driving test examiners. Part of the motorcycle test, as conducted in the late 60's & early 70's, involved the examiner watching the candidate ride at walking pace, by walking alongside him or her. Invariably, this was performed in Hamilton Lane, the road between the back of the stables and the Pub