There is an offline savegame bug introduced in 2.0 version of the game, and not fixed in 2.1. To be able to save your progress in campaign follow these instructions:

1) Launch the game and create an offline profile.
2) Exit the game and go to Documents\My Games\Unreal Tournament 3\UTGame\SaveData.
3) You will see a file named "Player.ue3profile" in the folder SaveData. 
4) Change the name of that file from "Player.ue3profile" to "<username>.ue3profile", where <username> is your desired profile name.
5) When you start the game now, enter your new profile name in the login screen, and choose "Offline Profile". Your profile name should be now visible on the bottom left corner of the screen.
6) Now progress will be saved and you will be able to continue campaign after restarting the game.