The miniature railway between the pier and the Dingle opened on 29th June, 1953. Originally steam, I think it closed for a period in the 1970s and reopened as diesel-hauled. As I remember the gauge was quite narrow (probably 7.25") and the engine was "Prince Charles". The engine was kept in a shed alongside the Dingle stream.
On the topic of miniature railways, Llandudno once boasted two. There was a short-lived one - two seasons in the early 1950s - on waste land at the rear of the Four Oaks Hotel, roughly where the swimming pool now is. The other was up at the Great Orme Summit during the time it was run as a Holiday Centre by Randolph Turpin and Leslie Salts. It is still possible to trace the line on the slope on the northern side of the Summit complex. The engine was "Commodore Vanderbilt" which apparently still exists in private ownership.