The answer is
I'm flushing away your poo and your pee?
Although it may sound so, that is not me!
The name sounds like water being flushed down the loo, flushing away poo and pee.
When people were fighting hard near to me,
It was not about who gets first to pee.
The Battle of Waterloo was about more important things than pissing.
The loser, who came from an island before,
Napoleon came from Corsica ...
Was sent to another one later therefore.
... and was exiled to Saint Helena after losing the Battle of Waterloo.
One winner came from an island as well.
Wellington, one of the victorious generals at Waterloo, was from Britain.
What place could I be? You sure know me well!
What cracked it for me was the 6th line. A loser being sent to an island immediately put me in mind of [redacted], and given that he also came from an island, I was sure it must be him. Then the first line fell into place, and I was ready to post an answer.