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For once TATY and I can agree on something!

Hmm. Maybe I speak snob English, but these are some ways I say that:

I speak English, and I am also proficient in Russian and German.

I speak English, and I can also speak Russian and German
...

I was of the opinion that уметь has to do with having a particular skill or ability, but not languages, e.g. я умею плавать. A search for "я умею говорить по-английски" gave me only 14 results, some...

I would not use уметь with speaking a language. The best is simply to say "я говорю по-русски".

Grammatically, I guess it is correct, but for some reason it just doesn't sound natural.

Не очень.

However, you double-posted.