What the photo/video doesn't show is that although they're not connected they are bodged with some kind of sealant, so normally it doesn't overflow, but in heavy rain it does. I think I basically have two problems: blocked downpipe and wonky gutters. Both should be fairly easy to fix though.
I'm just glad I've found the source. It doesn't show well in the video, but you can see where there mortar has degraded under that part of the gutter enough so that water can easily get in.
It was the blocked drain. I cleared it and it’s no longer spilling over in heavy rain. I’ve patched up the bad mortar. It was very crumbly in places under the gutter.
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