Works fine for me for several years now on Sid, and on my router.
I love nginx! Apache is way too much of a pig. It barely runs on plugs, and won't work at all on a router.
You need to be using php-fpm. You have to configure nginx and php before they will work together. It's not just a matter of installing the packages. There are countless tutorials out there on how to do it. The nginx site has good info.
It took me quite a bit of time to get it running the first time. But that was a long time ago when nobody was using nginx. Now it is much easier. The config is already there in the debian nginx defaults. You just need to uncomment the lines.