Both are, strictly speaking, correct. The placing of already in the sentence simply alters the emphasis:
John is here already.
The sentence above implies that the speaker is surprised that John has arrived so early. Compare it to the sentence below:
John is already here.
In this sentence, the emphasis is on here. It could be taken to indicate that the speaker expected John to be somewhere else.
In general, in English, the last word carries the greatest importance (source):
In written English, emphasis is largely a matter of controlling the way a sentence ends. The last words of English sentences carry the strongest degree of emphasis. When we maneuver into that sentence-final, stressed, emphatic position our most important ideas and information, we underscore the most significant idea through grammar. Even natural, intonational stress can seem weak and anticlimactic if we let a sentence end on lightweight words.
So, in your case, since you are emphasizing taken rather than already, I would go for
This username has already been taken by another user. Please pick another username
This username has already been taken. Please pick another username