I've tweaked the nginx configuration, and everything seems a bit faster.

Specifically:

• I've removed the line proxy_buffering off;, I thought it would send data back faster, but it broke the proxy cache. Outbound responses are now being cashed, so nginx can send e.g., JavaScript files nice and quick. I should, however, make sure that everything is sending sensible cache headers (i.e., private for generated pages for logged in people)

• I've added upstream stanzas for each server. This allows me to specify keepalive, to tell nginx to keep open the connection to then backend. I'm surprised it makes a difference on local servers, but apparently it does.

Next, although not very high priority, is to move all the nginx/backend traffic to Unix sockets, although I should check my belief that they're going to be faster first.