Turns out that the reason it was emptying itself of coolant was not a radiator failure at all. Apparently there’s a steel pipe that carries coolant from somewhere at the back of the engine to the front. That had corroded and split, spraying coolant through the radiator body from behind. It was a big split, too: about 50mm long
All fixed now and whilst a certain amount of stuff under the bonnet had to be removed, at least the front of the car didn’t have to come off as was the case when the alternator needed replacing (and I was fairly sure would be the case if the radiator had to be replaced).
Slightly concerning that a pipe carrying coolant is in a position where it can corrode all the way through, but I guess if it takes another 180,000 miles before it happens again, I shan’t be worrying about it.