I haven't heard of any normal isp services blocking p2p where you're situated. For dial up, i'm not sure upnp makes much difference. It's designed for dsl & cable modems which have internal NAT firewalls (block most incoming & occasionally some outgoing ports.) Dial up modems don't use NAT AFAIK anyway. BTW my brother used the same isp for a time. But he's now using adsl.

Which computer is the modem inside? If it's inside the desktop computer then you'll eventually need to get a hold of another modem. If it's inside the desktop computer, you can browse on the laptop right? But you suggested the laptop was the prime source so I presume that's the one with the modem. It sounds odd you can connect with the desktop & not the laptop if that's the case. You did configure the XP firewall right? Firewall configuration instructions (click on link) I would suggest to isolate the laptop whilst you're trouble-shooting this since that's the one you're later going to be relying on. It will also need maximum bandwidth to connect. Perhaps the networking is affecting it ... I don't know. But to fix things let's isolate the laptop for the time being (standalone.)

For the laptop, do steps 2 & 3 here Fixes for Limewire whilst LW is CLOSED. Then go thru the setup options again. Make sure it's set to modem connection, disable UP option under speed. Set the speed for minimum or around 5-7 KB/s. That's the max speed dialup offers. Give LW plenty of time to connect. It might take 20 mins, it might take an hour. You can also try setting the connection to manual connection. ie: Tools>Options>Connections & uncheck option to connect on startup. Then wait 5 mins, go to LW menu bar, File>Connect. As suggested, give it plenty of time. If no good after 20+ mins, disconnect (File>Disconnect), wait a min, try again.