http://www.yorkshireeveningpost.co.uk/news/full-steam-ahead-for-2016-as-world-s-oldest-working-railway-reopens-in-leeds-1-7819577
I thought the Tanfield Railway is the oldest. That one opened in 1725; Middleton opened in 1758.
According to the Wikipedia, Tanfield claims to be world's oldest, period, while Middleton claims to be the world's oldest continuously working public railway.
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