I think your current config is too complicated now so that your regular expressions do not do what you think they should.

Try a separate server setup in this manner:
server {
listen 80 default;
server_name .example.com;

set $project www;
set $user www-data;


if ($host ~* ^([^.]+)\.([^.]+)\.example\.com$) {
set $project $1;
set $user $2;
}

if (!-d /home/$user/sites/$project/public) { rewrite ^(.*)$ http://www.example.com permanent; }
root /home/$user/sites/$project/public;
index index.php index.htm index.html;
}

Create some dns names like user1.www.example.com (or just put them to your local hosts file) all pointing to the same host.
Create folders /home/user1/sites/www/public and put index.html stating "This is a home of user1" there.
Make sure you have access and error logs.

It should work.