A picture of my grandson reminded both me and my son of a book (two books at least in the series) by an English author that centered on a young boy who lived in the city with his mother. We both have vivid memories of him in his raincoat and putting on his galoshes. Everything about the book was quintessentially English – sort of Paddington mood. The boy lived in an urban setting, maybe working to middle class (not as posh as Paddington’s digs). The stories were about routine things – doing errands, the mail, etc. It’s driving us both crazy!
Maybe the Alfie books by Shirley Hughes?
This may be the books by English author Shirley Hughes called the Alfie books. They were a series about a little English boy. There is one called “Alfie’s Weather” You can see covers and all of the titles on Amazon.
If it’s a fairly old book, it could be Leila Berg’s ‘Little Pete Stories’: a collection of stories about the everyday adventures of a small boy living in the city. I think the cover illustration was of Pete out in the rain wearing his raincoat and galoshes.