How can I change a file's encoding with vim?

From the doc:

:write ++enc=utf-8 russian.txt

So you should be able to change the encoding as part of the write command.

Notice that there is a difference between

set encoding

and

set fileencoding

In the first case, you'll change the output encoding that is shown in the terminal. In the second case, you'll change the output encoding of the file that is written.

While using vim to do it is perfectly possible, why don't you simply use iconv? I mean - loading text editor just to do encoding conversion seems like using too big hammer for too small nail.

Just:

iconv -f utf-16 -t utf-8 file.xml > file.utf8.xml

And you're done.