Anyone who has ever tried to set up a UT server on a spare box on their home network has usually given up in frustration. It always works fine locally, but you never see it in the UT game browser. Feeling it's kind of useless running a server the world can't see, you shut it down in frustration.
Well, relax. The world can see it, even if you can't.
The problem is Network Address Translation (NAT), a worthless hack imposed upon the world in 1994 to save Internet Protocol version 4 (IPv4) from itself.
Your ISP has granted you one and only one globally unique IP address. If you have one and only one computer attached to the Internet, this arrangement works out just fine and your UT server will work the way you expect it to.
As soon as you start adding computers and other devices to your home network (a practice formerly frowned upon by many ISPs) you need to start leveraging NAT to share that single IP address. You have no other choice.
The first step is buying some sort of router and assigning the computers in your home (internal) network an RFC 1918 address space. Sometimes that address space (usually 192.168.0.0) is imposed upon you by the router manufacturer by default. The router assigns internal addresses to computers on your internal network and then performs Network Address Translation on the traffic to and from the Internet.
Basically, the router removes your internal address and replaces it with the "real" IP address alloted to you by your ISP for outbound connections. For inbound traffic it does exactly the reverse: it changes your "real" IP into the internal IP of the requesting computer.
This works extremely well for 99% of things people do on the Internet, and it also scales well, but it essentially breaks the way the Internet was originally designed to work, back in the day when 4 billion IP addresses seemed like an infinite resource.
When you publish a UT server through your router, it should work fine.
The reason you don't see it in the UT browser is that the IP address published to the world is seen as an external address to your home network. The only way to access that external address from your internal network is through the IP address your ISP assigned you.
UnfortuNATely, these two addresses are the same address.
You end up with a packet with the same source and destination addresses. This is judged to be an invalid packet by your router and it is silently dropped.
It's like calling your own phone number. You're simply not going to get through.
If you leave the server running, people will eventually start playing. You can play as well, but you must play through the address of the UT server on your internal network.
I work with intelligent, highly paid Cisco and Microsoft network engineers who can't, for the life of them, understand this problem, so don't let it bother you. Publish that server and give it some time.
If you build it, they will come.