Your paragraph about the “wal” element filled me with hope that it was the little shred of evidence that I was looking for, I’d never considered that before. I did a little digging around though and it seems to be that while the Germanic peoples did originally get this root from the name of a neighbouring influential Gallic tribe, it eventually became used to refer to the Romanised Gauls and then to any “Roman” foreigners, even the Italians. It looks like the “wal” you find in Wales, Cornwall, other place names and surnames was originally used to refer to the fact that these people were Romanised/Christianised, not because they were Celts. The English never referred to the Irish or the Picts with this name.

The French word “Galle” seems to come directly from this Germanic word too, so far from Pays de Galles (French for Wales) meaning “Land of the Celts” as we were all taught in French classes here in Wales, it looks like it simply means “Land of the Romans”, pretty much what Wales means itself. The English “Gaul” is a cognate of this.

The Irish “gael”, while looking and sounding similar, seems to have nothing to do with “wal/Gaul”, and neither does the Latin “Gallia”. The Latin comes instead from the name of a Gallic tribe, as does the Greek word where we get the modern “celt” from. The Greek word however eventually became used by them to refer to any non-Mediterranean European peoples including Germans, not just “Celts” as we know them.