Please check that router/software is not blocking ports 7770, 8089, 5672, 5222, 8080, 21020-21036, 7401-7600 for the game, also turn off all programs which use internet heavily (like torrent programs, netflix clients), disable antivirus, turn off k9 web protection filter or similar to that filters, also, you can try to put your router into DMZ mode
If this won't help, please send report through Steam Launch option "Report Connection Test" and post report name here (it will look like NAS-XXXX)
P.S.: You do not need to turn completly off your firewall and antivirus and other programs like that, if the issue is about them, you then need to make settings that allow you to play NA, how to change settings is usually desribed in program's help sections.