It depends on what you define a railway to be.  If it has rails and use flanged wheels, the Tanfield Railway may just be the answer.  Their website says:

“When the Tanfield Railway – or wagon way as it was known at the time – was built in 1725, it was a revelation. Its massive engineering was unlike anything else in its era, or even since the Roman Empire. It was a triumph of engineering over nature, a clear signal that a new industrial age was upon the world, and that railways would play a massive part.

First laid down more than a quarter of a century before the first railway officially sanctioned by government, over 75 years before the first steam locomotive and a whole 100 years earlier than the Stockton and Darlington Railway, the Tanfield Railway is the world’s oldest railway. We will be the first railway to celebrate our tricentenary in 2025.”

The railway also includes the Causey Arch, the oldest surviving railway bridge in the world. Click here for more of their history at their website and click here for the Wikipedia entry.

In the video above, we see a special event on this tourist railway which featured a vintage freight train in action for the benefit of visiting photographers.

Click on each image below for a closer look!

Causey Arch