I can't speak for Birmingham - the only delivery people I ever saw were the regular milk, bread, coal. beer and mineral water vehicles. Some of these post ww2 were still horse drawn but many slowly gave way to motor vehicles or ceased altogether.
In smaller towns there were other delivery people such as fish carts - usually straight from the local boats/fish market - paraffin sellers, fruit and veg carts and so on. These were often a boon to those who could not walk far or were otherwise housebound. If people had cars and not many did until the early sixties, they would most likely be used by the man of the house to get to his place of work or do his job.
With the post war expansion of housing - even in small places - non motorised vendors could only reach a small number of their customers. The cause for their customer loss was that many town centre houses became shops and the occupants moved out to the new residential areas.
Now it seems that many of those shops have gone and the premises are dwellings once more. I guess Covid could increase that happening.