Starting in FateUHC V3.81 you can now disable world generation and use a backup system to use the same world each game.
Head to your /plugins/FateUHC folder and find the folder inside called backUp
Enter this folder and put all your world files.
Head to the uhcconfiguration.yml and change
Then you need to start and stop the server once. It may send errors the first time you do this.
The next time you start the server you should be good to go!