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. 

How about Frank as a "Rag and Bone" man?

Here's a photo of one in London in 1986:

https://www.mylondon.news/news/nostalgia/gallery/incredible-pictures-show-londons-last-23173933.amp

https://en.wikipedia.org/wiki/Rag-and-bone_man

I think Frank could be a rag and bone man, or a milkman. Horses were used to pull milk carts up to the 1980s in parts of the UK. So either would work well for the era.

The horse in the set photos looks much more like a horse that would be pulling a cart rather than one that would pull a hansom cab.

Here's an image from Edinburgh in the 50s:

I did wonder initially about a royal parade / coronation tie-in but as mentioned by Starkat the harness isn't right for that at all. There's no saddle for a horse pulling a cart, but it's possible the saddle will only be there for the stunt rider shots?

Where I live there are a lot of horses, and within the last couple of years I've seen someone drive a cart to the pub and then stand outside holding the reins while drinking their pint. So even in 2025, perhaps ...

Oddly enough, in London this weekend I saw no less than 15 horses and traps driven by Gypsy / Traveler Folk in two separate groups along Haymarket and on Birdcage Walk.

There were also tonnes of police horses out doing crowd control for a large protest taking place along Whitehall... Not to mention the two horses that were on set for filming!

Needless to say, conjuring a working horse in London ought not to be a difficult task for any screenwriter, no matter what era it's set in

@icarus I'd prefer that Frank is a milkman, rather than a rag-and-bone man. It's classier. (did you ever see Steptoe and Son? scruffy, grubby old bloke and his son who wants to be a more elegant chap)

I hope Frank has still come from the countryside, where he and Strawberry worked in some farm work or perhaps carting. 

@narnian78 I often have wondered why C. S. Lewis renamed the horse Strawberry Fledge.  Apparently it has to do with the fact that the horse actually learns to fly like a bird leaving the nest. And Fledge may come from the word fledgling, which refers to young birds which are learning to fly.

Birds are either hatchlings or chicks once they hatch out of their shells. When they grow big enough to start growing feathers but aren't yet fully FLEDGED, they are called FLOOFS. We were watching on You Tube, an ornithological study of a pair of mating peregrine falcons at the Orange, NSW, campus of Charles Sturt University. That is why I like your suggestion that Strawberry/Fledge's name did come from Fledgling - literally in his case. 

@icarus How about Frank as a "Rag and Bone" man?

I don't know how old that picture would be, or which part of London it is, but I have to admit that the terraced houses look exactly right. Out in the countryside in UK, near Sandringham or near Windsor Castle, or at Cirencester, is where you'd find much horse breeding for races, polo, army horses etc., wouldn't you?

I wonder if C. S. Lewis actually believed that there were flying horses.  It seems that he might have believed they were real and may have once existed in our world. He had a fascination for classical mythology which may have extended into reality. We don’t have anything to prove the existence of flying horses except that a constellation, Pegasus, was named after the one in Greek mythology, and they are sometimes shown in artwork.  But I don’t think that minimal evidence would have stopped Lewis from believing in them.  And there probably were some mysterious creatures from the classical mythology that really existed.  🙂