It´s such a core strategic issue, that EU ought to set aside billions of EUR a year for this kind of stuff. Not just an OS, but a complete productivity and technical suite that can cover the vast majority of functions that are needed at workplaces.
The fastest path is of course not to start from scratch, but take what is and get it up to a level where it´s usability matches professional software, both in regards to UX and stability. If I was EU I would make contracts with existing projects funding/services for functionality, so the EU can set aside an agreed amount of money that are released at certain milestones and even offer professional developers/pogrammers from a pool of their own.
No matter what, EU is forced to do something like it, based on the current circumstances. It´s absolutely essential for EU security that EU starts to develop some form of software independence, like China and Russia does. US holds enormous leverage over EU as it is now and EU buys large amounts of licenses from the US. They have all the incentives to change this.
I don´t know if you remember US Echelon spying on EU companies in the end of the 90's? I really can´t comprehend how the EU has not already started a project for software independence long time ago.