A few passengers services were horse drawn in the past and one (though really a tourist tram) to this day

These first from Wikipedia:

Fintona

The Londonderry and Enniskillen Railway opened the station on 5 June 1853. From 1856, mainline services were withdrawn, and the station was a branch line from Fintona Junction railway station. Most passenger services on this branch line were provided by a horse-drawn tram car.

The short branch to Fintona became famous as the GNR worked it with a horse tram. Since the line's closure, the tram has been preserved at the Ulster Folk and Transport Museum at Cultra, County Down.

Port Carlisle

As a cheap alternative a horse-drawn service was provided in 1856 between Drumburgh and Port Carlisle for a number of years. In 1914 steam power was used and to try to avoid closure, a steam railmotor called 'Flower of Yarrow' was built and this service to Port Carlisle railway station lasted until the branch was closed in 1932.

Douglas

The Douglas Bay Horse Tramway on the Isle of Man runs along the seafront promenade for approximately 1.6 miles (2.6 km), from the southern terminus at the Victoria Pier, adjacent to the Isle of Man Sea Terminal, to Derby Castle station, the southern terminus of the Manx Electric Railway, where the workshops and sheds are located. It is a distinctive tourist attraction. However works have been underway to relay all of the track in 2019 and at present only a third (from the Derby Castle) is usable, with no published completion date for the works.

York - from History of York

For the first few months a steam tram was trialled, but soon horses were pulling the trams.  The line was extended through Tower Street, Clifford Street, Ouse Bridge and Micklegate to the Mount by 1882, with a branch along Rougier Street to connect with the new railway station.

Micklegate is a fairly steep incline and an extra horse was needed to pull up the hill.  A horse called Dobbin did the job for many years, waiting at the bottom of the hill until the next tram came along.  He would be hooked up, do his work and then be released to make his own way back down to his position outside the Post Office.

No reason ... other than that I started putting the OTD^▸ (On This Day) data set together with the slightly unusual, then searched on the dates that were missing to see what I could dig out in the more conventional.   Guess Charlie came out tops!

I will add Trevithick to my data set ...

Paddington Station had the Mint Stables https://historicengland.org.uk/services-skills/education/educational-images/dray-horses-at-the-mint-stables-paddington-station-westminster-london-3921

Now part of St Mary's Hospital called Mint Wing? They're remnants from the days when the building housed multi storey stables for the horses that worked on the railways at Paddington — up to 600 horses at any one time.

The building remained as stables until the 1950s when it became a research laboratory.

The horses there would have been used for delivery drays but would have also stabled some of the 'shunting horses' used in Paddington Goods

From 1858 to 1882 the Bridport Railway had to haul the passenger trains the last few yards by horses! There was no run round loop and there or at Maiden Newton. The latter had a gravity siding  to shunt the train. Here the train backed up the siding, the engine ran into the station and onto the branch while the guard brought the train down the siding into the bay.
At Bridport, the train stopped outside the station and the engine was uncoupled and taken to the shed. The shunting horses were attached and drew the train into the platform.

In 1882 the platform line was extended a few yards, they could not afford that before. Now they had the powers build to the harbour and thus could raise the money to pay for this extension. But not before looking at building a new terminus station where East Road crossed the line.
The final decision to build to the harbour was taken shortly after.

The following year saw the contractor’s engine running in steam from the station to the harbour. In steam and with 6 horses to guide it down Barrack Street, East Street, South Street and Harbour Road! Shades of the Titfield Thunderbolt!