So finally got it all right. As mentioned in my comment before (Issues with no tty present, when using Jenkins), the issue wasn't directly the call itself. The issue what that jenkins was asked for password, which didn't appear in output.
Solution was to first correct sudoers, and then make a small change to call (replace nobody:nogroup with nobody.nogroup):
ssh -i {$id_rsa} {$user}@{$server} sudo chown -R nobody.nogroup {$root}/{$checkout_path}/
So actual call would be something like this:
ssh -i /var/opt/jenkins/.ssh/id_rsa -t jenkins@remote.server sudo chown -R nobody.nogroup /var/www/some/directory
Sudoers now looks something like:
jenkins ALL=(ALL) NOPASSWD: /bin/chown nobody.nogroup *,/bin/chown -R nobody.nogroup *
Ps. I'm aware of the security issues with letting jenkins user use sudo chown nobody.nogroup anywhere. On the other hand it wouldn't do any good to restrict with /var/www/* as it is very easy to get around that anyway. So for security reasons it would probably be better to do something completely different...