For the curve between Hindley South and the Lancashire Union (Whelley) line at Amberswood West Jn, I agree that coal was most probable, with possibly occasional local mixed freight. I doubt this curve ever saw any regular medium or longer distance goods traffic.
The original construction of the Wigan Junction Railway to the location which eventually became Hindley South, and its connections to LNWR lines in the area, was heavily wrapped up with the interests of local coal mine owners. According to Dennis Sweeney's book The Wigan Junction Railway, the earthworks & trackbed for the Amberswood South curve were constructed by Crompton & Shawcross (owners of the adjacent Strangeways Hall colliery) with the track being laid by the Wigan Junction Company. Ordnance Survey maps dated 1888 and 1907 reproduced in Sweeney's book show significant sidings and connections from this curve to the Stangeways Hall colliery and onwards to the network of private industrial railways in the Amberswood, Ince and Hindley coalfield.
Another of Sweeney's publications, The Lancashire Union Railway mentions that in the early 20th century, local mine operators had agreements with the LNWR and the Great Central to run their colliery locomotives on the lines around Amberswood West Jn, Amberswood East Jn and Hindley South to access their various mines in the area.
This all points to coal traffic being of highest importance.
As for other goods traffic over the Hindley South/Amberswood West curve, I have access to a WTT and a list of train movements in the area, both from the 1950s. Neither show any movements over this curve.
I've also seen a photograph dated around 1960 taken at Hindley South station, showing that by this time the junction signal for the route to Amberswood West had a small semaphore arm, compared to full size arms for the Wigan Central and Amberswood East (for Standish) routes. This suggests the Amberswood South curve had been downgraded to "through siding" status. Not surprising, since most of the smaller, shallow coal mines which had been the motivation for Wigan's dense network of colliery railways had become exhausted or uneconomic by the period between World Wars 1 and 2.
If I had to hazard a guess (although I have no primary evidence to demonstrate this) I'd say much of the traffic would have been local trip workings (not specifically timed in WTTs) between local collierys and the sorting sidings at Bamfurlong and Ince Moss. Bamfurlong Sorting Sidings was a busy place, specifically built by the LNWR for re-marshalling of local goods trains for onward dispatch, and in the opposite direction for sorting empty private wagons for return to their owners.
As Bevan Price has commented, the section of the Lancashire Union line between Amberswood, Ince Moss and Bryn carried quite varied traffic, including heavy longer-distance freights such as the famous Long Meg-Widnes anhydrite trains (which were often photographed on the S&C in later days of steam, but strangely not very often in the Wigan Alps!)