They certainly spoke a Celtic language, but that did not necessarily make them Celtic.

In point of fact, it does. When historians and anthropologists refer to groups of people as Celtic, Germanic, Italic, etc., they're making a primarily linguistic distinction. These people, who spoke related but differentiated languages (there were five distinct varieties of Germanic by the early centuries AD), would likely not have understood themselves to be part of a common group in any meaningful sense. We group them together because of common language, not because of self-identification.

The video compares this to modern day Britain, just because English is a Germanic language does not make them German.

This is a really misleading comparison. Germanic and German have entirely different meanings; the one refers to language, and the other to nationality. The English very arguably are a Germanic people, in the sense that they speak a Germanic language.

And if there is, what ethnic group did they belong to?

Basically, I don't know that this would even be applicable to them. Ethnicity is more of a modern concept that doesn't work all that well when applied retroactively. In all likelihood, they would identify with their local political group: clan or tribe or people or whatever you want to call it. Pre-Roman Britain was not a unified place with a common ethnicity.