Photo credit: : National Trust, Hartwell House
The castle on the far hill may be identifiable. Could this be No. 366 ‘A View of the Waterfall of Llangollen and the Mountain Called Moel-y-Bache’, one of the two paintings which the artist exhibited at the Royal Academy in 1773? A painting of similar size and style, ‘Landscape with a Fishing Party by a Waterfall’, is in the Ashmolean Museum of Art and Archaeology. No. 365 at the same exhibition was ‘A View of Llam Leids on Dee, a Mile beyond Llangollen, North Wales’. Waterfalls were Mullins’ exhibits No. 134 in 1771 and No. 162 in 1772 as well.
The hilltop castle could be the impressive ruined Castell Dinas Bran of which there are many photographs on https://bit.ly/2DDr9WI
The ruined castle on the far hill looks like Castell Dinas Bran. But I think that you may have already worked this out. Given the likely scale, is it not possible to consider modern topography even from an OS map to identify a waterfall? The River Dee has changed much since 1773 with the construction of the canal, and the most noted waterfall appears to be the Horseshoe Falls. This is more of a weir designed by Thomas Telford to draw water into the canal from the River Dee. Artistic licence might mean that Pistyll Rhaeadr, one of the Seven Wonders of Wales is depicted. The latter falls are located not too far away from the RiverDee. But I would conclude that this is an outside guess.
Hackney Museum's late 18th century view of the Pistyll Rhaeadr described there as 'A Rocky landscape with a waterfall and an artist seated in the foreground' shows just how impressive those falls are compared with this waterfall. That painting is attributed to Jacob More. David Solkin has suggested that it may be by John Inigo Richards . Another possibility is that it is by Michael Angelo Rooker.
Although there has been quite a bit on that work under 'proposed discussions', it has not yet been accepted for open discussion
Rheidol Falls is a possible candidate, with some romantic interpretation of the background.
Martin, thanks for drawing my attention to 'A Rocky landscape with a waterfall and an artist seated in the foreground'. I'll follow this up with the collection.
Contrary to the suggestions already made, there is not much about the topographical profile of this scene that bears any resemblance to the conical hill upon which sits the castle of Dinas Bran, nor to the very widely represented architectural depiction of the castle itself. It certainly bears no comparison to other 18th century paintings and engravings of the site.
A Google Images search for "Dinas Bran" offers a multitude of different images, both contemporary and modern, from a multitude of positions, none of which look like the scene presented in this painting.
In 1967, Christie, Manson and Woods sold "The Falls of Llangollen, with shepherds and shepherdesses and animals and Moel-y-Bach in the distance, signed and dated 1772, 30" x 40&1/2"" (Our painting is measured as approximately 40" x 50").
If this was the work shown at the RA in 1773 as ‘A view of the waterfall of Llangollen and the mountain called Moel-y-Bache’ and as exhibit number number 366, then it is not this discussion's painting, especially as, apart from the dog, there are no apparent animals in the painting, as neither are there shepherds and shepherdesses.
Further to Hackney Museum's late-eighteenth-century view of the Pistyll Rhaeadr, described there as 'A Rocky landscape with a waterfall and an artist seated in the foreground', the collection is in the process of appointing a member of staff who will be our new Art Detective contact, so we should be able to open that discussion in a couple of months. They need to look at the painting first and there is limited access.
Along the river or lake side immediately behind the waterfall is what appears to be a Weeping Willow. Research suggests that Weeping Willows were all grown from cuttings of a single tree first grown in Twickenham in England in the 1740's - https://stmargarets.london/archives/2010/08/the_weeping_willow_tree.html and also on Wikipedia.
If that is the case the trees on the edge of the water are more reprentative of a parkland or garden setting and not a wild landscape in Wales.
It is an intriguing possibility that this painting might show a Weeping Willow tree that, if this is one of Mullins' Welsh works, could be thirty years or less in age. However, there is also the possibility that it is a Weeping Birch (Betula pendular...), a tree native to England and one that was described in 'The Journeys of Sir Richard Colt Hoare through Wales and England, 1793–1810', and one that was depicted in 1812 in John Laporte's engraving of 'Weeping Birch, in the Vale of Llangollen', a copy of which is in the National Library of Wales.
See https://bit.ly/2C8TQZk for a description of the Birch tree.