The exact location is right under the bridge on the Abbey side. From the A66 turn off at Rokeby Park for Barnard Castle, follow the road until you get to Egglestone Abbey (visible) and park just after the bridge lights. Walk to the bridge, to the right of the bridge follow the leafy path down the slope directly to the river.
You basically jump or dive in and scramble out.
The water is peaty and dark making the depth impossible to see, also the riverbank is blocky. Having said that, the drop is over 2m and the depth is around 2m to 2.5m (I measured the depth with a rod and line). I have jumped and dived in from both sides and have touched nothing! The deepest section is towards the right side of the drop (downstream) at its highest point, where there is a current. The current is mild and the water is warm during sunny/dry spells. When you are done swimming just make your way downstream from the jump and scramble up the blocks. Take care when picking your way up the rocks. Where you jump in, there is a subterranean overhang where your legs touch nothing! So it seems to have good clearance! All the same – stay clear from the rocks when jumping in.
Sorry to ramble on!