Something to research. Where did it start? Where was it going?
Was is for tourism? locals? or industry? or for WW1 or 2?
References:
“Milford remained a small mainly agricultural settlement of some 50 or so people until the last decades of the 19th Century, when Colonel Cornwallis-West of Newlands Manor aimed to make Milford a fashionable resort (called Milford-on-Sea) complete with pier, railway station, public baths, hydropathic establishment and a golf links. The plans failed primarily due to lack of capital and of market interest. ”
(https://www.newforest.gov.uk/media/adobe/MilfordDesignGuide.pdf)
There was a narrow gauge railway built to service Hurst Castle, but records don’t fit with size or destination.
“A new public house, the Castle Inn, was established on the north end of the spit.[69] A narrow-gauge railway was constructed in the 1880s to move supplies into the castle.”
(https://en.wikipedia.org/wiki/Hurst_Castle)