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I would be looking at just removing any render and cement mortar between the brick/rubble and repoint in with lime mortar at a radio of 5 sand: 2 NHL2.5. possibly color match it if you CBA but not all dyes work with lime. The water is always going to get in if it's below the DPC so it'll blow off anything non breathable as you've found (also kill of all the plants first obvs). The tricky bit is mixing. It can be done by hand, but keep going longer than you think and don't add to much water at once. The mix should look quite dry at first but will get more plastic the more you work it. Rake out the joints to about 25mm before hand with a range of small cold chisels like these and then press in the new mortar with something like these or these. Wait a few hours, brush the loose bits off once it's gone off a bit
Oops, that was meant to be @lemonade
Update. There does appear to be a thin layer of slate between the area that is missing its render and the house bricks proper.
So does that mean if this area is below the damp proof course I don't need to worry about water penetrating?