Regular, scheduled passenger services certainly ran between Southport Chapel Street and Ormskirk via the west-to-south curve at Burscough. They continued until the early 1960s.
However, I don't think the west-to-north curve ever carried regular, scheduled passenger services. Trains between Southport and Preston or East Lancashire always ran via the former West Lancashire Railway route via Crossens. The curve may have been used for excursions or special holiday trains during Wakes Weeks, but other than that I believe it was only used for freight.
I think you're right on this. The last regular working over the North Curve was a Blackpool-Southport ECS working, on, I think, Saturdays only. There was also a light loco working, of which more below. The North Curve was closed and severed at the Rufford end in 1969, and the Burscough Bridge end in 1971, though the track stayed down until into the mid 1970s. See here.
https://www.flickr.com/photos/63333186@N02/19317309820
Your right it never did, the south curve stayed in use for connection to the RO munitions factory at Burscough and I believe, a regular newspaper train. but the land the curves are sited on is still owned by NR.
South Curve stayed in use until about 1981, though it was physically disconnected from the Ormskirk line after summer 1970: it crossed over a flat diamond crossing.
The newspaper train you allude to was a very early morning Manchester-Southport turn that was, incidentally, the only working to ever be diagrammed for a class 50 on the Wallgate-Southport line. The loco would detach at Southport (think it ran into the old Southport Central station, then still in use as a goods shed) and then return to Burscough Bridge. It ran up the North Curve, crossed over there, and then ran down to Kirkdale carriage sidings to collect the stock for the 9AM Liverpool Exchange - Glasgow working.
Obviously this will have ceased in the period between July 1969 and May 1970, when the loco-hauled expresses ran out of Exchange but the North Curve was severed. The locomotive presumably either ran to Liverpool via the coast line and turned at Sandhills, or else via the South Curve. I'm sure I remember reading somewhere that by this point, Burscough Junction South, a draughty wooden box, was mostly switched out, so perhaps the coast line was used.