What happens with RNAV procedures on the way from AIP publication to FMS database?
The navigation database is technically a component of the GPS or FMS navigation system.  As such the manufacturer of the GPS or FMS is responsible for ensuring the database works with the equipment.  


The database itself is usually prepared by a database provider such as Jeppesen. They have to work with the manufacturer to format and structure the database to work with the GPS or FMS.  After Jeppesen builds a database for a given cycle the GPS/FMS manufacturer will test the database before it's released.   

Depending on their business agreement, the database subscription may be purchased from the manufacturer or direct from Jeppesen.

Jeppesen obtains the data from the ANSP and must process the data in accordance with RTCA DO-200B, Standards for Processing Aeronautical Data.  (Available for purchase from RTCA Inc.)  


In the case of the FAA, the data is available in numerous formats, most commonly in ARINC 424 format.  


There's a great deal of info available on the FAA Aeronautical Information Services website.