There is a lane (Lon Cefn Ty) that leads from Lon Pobty and comes out near the roundabout by the Castle Gate at Llandegai.
Where this lane crosses Afon Cegin is a ford which looks remarkably like the postcard.
That google car has been down that way so you can have a look.
Oldmaps has Bishops Mill at where Bishop Mill's Road road meets Afon Cegin. In between the Mill and the ford is a Weir. About a 1000 yards from ford to Mill.
Having said that. On google maps you can see the line where Bishop Mill Road at MaesG would run towards the river (and the industrial estate opposite).
If you drag the little man over towards the river you'll that some one has posted some photos at the river crossing. Oldmaps shows that the continuation of Bishops Mill Road over the river is a short footpath whereas the postcard suggests a road either side of the river.
In the postcard that fencing either side of the gate looks to me like the slate upright type of fencing. If you position yourself at the waters edge at the Ford on Lon Cefn Ty and look at the entrance to that little car park you can see a similar fence has been left among newer fencing.