How do I get netcat to accept connections from outside the LAN?

You must listen on the public interface. You're currently listening on localhost where no one outside of your computer can connect.

Your milage may vary, but with my implementation of netcat I have to use this command line to listen on all the public interfaces.

netcat -vv -l 0.0.0.0 6666

To specify a listening address:

nc -l -s <LISTENING_IP_ADDR> -p <LISTENING_PORT>

Tested on a Debian Jessie with netcat-traditional-1.10-41

Turns out there is no problem. It only appears that way because my router doesn't allow hairpin connections. That is, even though I've got it set up correctly, the router wouldn't make the connection when both source and destination are behind the NAT. Simply ncat -l -p 6666 works fine, so long as the request comes from outside the LAN. To test this I browsed to my.external.ip.address:6666 with my 3G mobile phone and sure enough, a HTTP request came through :)

This answer came from: a serverfault question, which is where I should have asked this question in the first place. Apologies for that.