I use the JA+ mod. Whenever I play a LAN game, when the time/frag limit is hit and the map changes, the mod doesn't load on the next map. Whats the deal here?

Unless it's a problem with the mod itself (in which case you should talk to the author), the map should cycle if g_automapcycle 1 is set (check your cfg file for exact details).

Note that in order for the map to change, a human player (bots won't do it) has to be physically present on the server and hit "ready" at the end of the map.

This is a problem in ALL Q3 engine games as far as I know, unless somebody is there to hit "Ready" the map will never change, EVER. It's really stupid and annoying, but that's how it works.

Of course if you're the host of the game you can always force the next map (using the map <mapname> command) from the console, but if you are running a dedicated server it's annoying to have to be there all the time to monitor it and make sure the maps cycle. Since often people will wait for their favorite map.