What we call Celtic knotwork design isn't uniquely Celtic. It's probably not even originally Celtic. The Celts, who first showed their tattooed faces in central Europe a very long time ago, loved bright colors, curved shapes, wine, and mayhem. They didn't write anything down - like a lot of tribal peoples, they believed that writing was a magical act that you shouldn't undertake casually. It's because by setting the name of a thing down as an object you have created something that has power over that thing. Think voodoo dolls: you're on the right track.
But they didn't let their respect for writing stop them from making their mark. They decorated everything. Everything. The early Celts picked up design influences from the Scythians and Greeks, mainly, and learned some new tricks as they spread west across Europe, over the English Channel, and beyond the Irish Sea. But they weren't doing knotwork designs yet.
Ancient Celtic art, especially in the La Tene style, is all about curves and color. It's gorgeous stuff, but it's not what we're looking at today.
Knotwork or interlace design is a lot more like ancient Saxon and Scandinavian art, which is full of twisting gripping beasts and interwoven lines. It's also great stuff, but on its own it wasn't quite there yet.
By the sixth century the Celts had become pretty well settled in the British Isles. They were the Britons, after all. The Romans came, made a lot of roads, whipped a lot of people, then left. The Britons might have had a chance to figure out who they were again, but that's right about when the Saxons showed up.
In most cases when one group of people conquers another one they turn the old people into peasants or slaves and get on with it. You can tell a lot about how this works by looking at the place names in a country. Think about all the Native American names for towns, rivers, and mountains that we see in America. The conquerors ask the natives, "What do you call this?" It's like that.