ABSOLUTELY UNBELIEVABLE!!
8 hours in and the only clue from the multi-meter being voltage on the headlamp washer pump pin that shouldn't be there. But as we have no headlamp washer system it was ignored right up until desperation struck. So what, in the course of changing the loom, might we have wired up wrongly to make the wiper module think we had a headlamp washer system? Just in case that fixed it...
The clue is here. The washer pump part of the wiper module connectivity. The black line on the left. I ignored it the first 99 times I looked at it then realised it was possible it was a jumper. So off I went to the known good car and what do we find? A jumper cap on the end of the headlamp washer pump connector... Sure enough, swapping it onto Willy it fixed the issue. And just for good measure we looked for it on the old burnt out loom and found it.
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So there you go. BMW supply everything ready to go with a new loom except for this stupid jumper cap (61138380192) that feeds a ground into the wiper module. It really would be sensible to include it because if you do have a headlamp washer system you'd remove it to fit to the pump. If you haven't got the system then, well this situation is highly likely to result! To the best of my knowledge I can't think of anywhere else on the loom that uses a jumper for a non-present fitting. So it just didn't occur to me. Bizarre.
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