Bereit has the implication of being prepared (to do something). For example, you could have prepared yourself for a marathon (training etc.) so you can then say:
Ich bin bereit für den Marathon.
You can also prepare stuff, the corresponding verb is bereiten and they share the same etymological route. E.g:
Ich bereite das Essen.
Das Essen ist (zu)bereitet. (stative passive or adjectivised participle)
Das Essen ist bereit (und kann gegessen werden). (using the actual adjective bereit)
The word fertig on the other hand means finished. Finished can mean ready for the next step, but in general the idea is that something recently has been completed. Taking the marathon example, you are bereit if you have completed your training plan and eaten lots of carbohydrates the night before. Then you go to the actual place, put on your gear, tie your running shoes and do warm-ups. Now you are fertig with your direct preparations and ready to start running.
Ich bin (mit den Vorbereitungen) fertig. Es kann losgehen.
Similarly for the food example, using fertig means that the cooking is complete. It is now ready to be eaten.
Das Essen ist fertig.
In your paperwork example, it depends on where you want to place the emphasis. Are you looking backwards once the paperwork is ready, i.e. has it been completed? Then you should use fertig.
Wenn der Papierkram fertig ist, überweise ich das Geld.
If, on the other hand, you wish to imply that the paperwork needs to go through a few more hands before the message’s recipient can enter their data, you will need bereit.
Wenn der Papierkram bereit ist, können Sie Ihre Kontonummer eintragen, und dann überweise ich Ihnen das Geld.
From the way you phrased it, fertig sounds like the better choice.