Your licence is issued by the State that issues your medical, that is up to you where you do it. Doing the ATPL exams has nothing to do with your medical or State of licence issue. If you simply want them for a FI rating then any EASA State is OK as they are just a demonstration of CPL level knowledge. If you want to instruct in UK just do the FI Course and have it added to your UK issued licence, no need to change anything. I wondered where Austro was, Australia? Austria?
If you ever want to teach commercial then you will need a CPL, your ATPL exams will expire after 3 years for licence or IR issue but they remain valid for a FI rating.