Both your sentences are correct, but 'undisturbedly' would be wrong.
This is because the word 'undisturbed' does not relate directly to the way that you're studying - you can study hard [yes, it's an adverb] or you can study carefully, for example, but you cannot study 'undisturbedly'.
When we say "...I can study English every day undisturbed", this is an elliptical way of saying '...under conditions in which I will not be disturbed.' 'Disturbed' is a past participle, used as an adjective. It refers to your state, not to the manner in which you are doing something. This is why the adjective is correct and the adverb is not.