That says to me that the reason you can access your server and websites is because you are 'contacting' them via the DNS entries on your own network - you aren't going outside your own network. I'm guessing therefore, that WAN access (internet) to your websites is via port forward on your router, and as the WAN appears to be down, you won't be able to get to them UNLESS you're on your own network. You can check this by using your phone (iPhone?); turn off the phone's WiFi and just connect to the internet via mobile data. If you use the phone's web browser and put in the website address (like www.google.com), you'll probably find that you can't get a response - that will confirm that the WAN is down.

Next question; presumably your router provides local IP addresses(?) - or is the DHCP server held on on the server? - in my case, I have 2 standalone servers, with a router (static local IP address) acting as the gateway to the internet (via cable modem). If the DHCP server is on the server, make sure the router has a static address so the server knows where/how to get to the WAN - which will be the gateway address you've defined in the DHCP server.

Next: is the WAN (internet) via a cable modem or is it an ADSL modem? (ADSL being a glorified dial-up). The difference is that a cable modem connection is there all the time, and will be there assuming the ISP side of things is up and you've paid your bills . If it's ADSL, then the connection can either be 'on demand' or 'permanent' - that's a setting in the ADSL router, and it would need to be set to the 'permanent connection' if you want your webserver to be visible from the web all the time.