How does an American write this?

That will be perfect timing. When you're done I'll have finished as well.
The way you have it written there is perfectly fine.
Hi,

That will be perfect timing. When you're done, I'll finish it as well.

I don't think the future perfect tense fits this context; it would if the sentence were 'By the time you're done,

I'll have finished as well'. Of course, the meaning changes then.

When indicates exactly at the time you're done, not before it.

Regards