The title says it all! Even if Anglo doesn't quite mean "of the English" you get what I mean.
Cambro- is the traditional form; the OED defines it in part as:
…used as combining form in the sense ‘pertaining to Cambria, Welsh’.
It’s never been common, but has clung on tenaciously over the years, in cambrophone, cambro-centric, Cambro-American, and the like. (It’s much more common in geological use, where it refers to the Cambrian Period.)
However, few apart from classicists or historians will understand it. An alternative option is Cymru- or Cymro-. This is less traditional, but usage on the internet and in British newspapers suggests it’s probably about as common today; and it has the great advantage that most Brits, and certainly anyone who’s lived in or near Wales, will understand it immediately. So if you don’t mind creating etymological Frankensteins, this is what I’d recommend.
In my opinion, Kate Gregory is on the right track.
The Latin word for Wales is Cambria, so by analogy, the prefix would be cambrio- - Cambriophonic, Cambriophilia, Cambriocentric, and so on.
When choosing a prefix to attach to an existing word in order to make a new word (a neologism), the convention is to use a prefix with the same linguistic root as the main word. Cymru- would be inappropriate for prefixing to words with Latin origin, or that come to us in English through Latin (as most Greek roots do). Cymric would be better, it and is certainly the English-language adjectival form of the Welsh demonym, but it doesn't have the form of a prefix.
Of course, if you want an adjective that stands as its own word, Welsh or Cymric is best, though frankly, Cymric isn't as widely used.
So you might want to say a "Welsh speaker," "Cymric speaker," or a "cambriophone" depending on your audience.