This question was an excellent excuse to spend an hour or two browsing through books on the history of the area.
I was intrigued by several historic Ordnance Survey maps (e.g.
this one) showing a minor railway line terminating close to the brewery at Guest Street.
I wondered if the Leigh Brewery had had a similar arrangement to
Magee Marshall's in Bolton (at the other end of the Bolton & Leigh Railway) where a railway siding ran directly into the brewery premises and was used for delivery of rail tankers of water from Burton upon Trent. Local legend had it that a tankard was permanently chained in place by the siding, allowing the railway crews making these deliveries to sample Magee's product before proceeding to the rest of their day's shunting work around Bolton.
In the case of George Shaw's Leigh brewery, it seems the nearby railway was a private colliery line, used only to connect the Bedford Colliery just to the north with a landsale yard in Guest Street - basically where the local coalmen would go with their horse & cart to pick up sacks of domestic coal for delivery around the town. This private line did connect with the main L&NWR line at Speakman's Sidings between Leigh and Tyldesley, but it would only have been used for the colliery's own coal traffic.
This means that incoming raw materials and outgoing "products" from the Leigh Brewery probably went via the town's general goods yard adjacent to Leigh station which opened in 1864 (a couple of years before the brewery) and replaced an earlier goods yard located at Westleigh station on the older Bolton & Leigh line. However I've not found any reference specifically to the brewery's activities and rail transport.