There were still some horses and carts used in English towns and cities in the 50s. Cars were expensive, and everyone got about by bicycles or on foot, or by bus or train if going further. 
A TV series about two rag-&-bone men was popular in the early 60s, (early recycling!), used a horse pulling a cart. Stories and songs often involved horses and carts.

(I can remember being called to run to the front window to see an old man on a nearby farm occasionally driving his horse and cart down the road past our street, very rare, about 1960, in New Zealand).

Would carriage horses have saddles equipped when pulling carriages, or is this here to make riding it easier?

Carriage horses are still present in major cities to give leisurely rides in modern times, so I don't think having one here, even if not a hansom carriage horse, would be too unrealistic.

The last hansom can in London finished in 1949. I did look it up. In London, with the cars, carriage horses would be restricted to certain areas of town. They wouldn't be allowed outside of them. 

I vaguely remember seeing a mounted rider in a royal parade of some kind. I honestly think it was the coronation, but they are missing massive pieces of the harness.