I am not familiar with verification procedures for UK residents. However, to answer your specific question ("will I need a simple check, or do I have to do some kind of test?"), I believe the answer is: it depends upon what sort of Australian license you would like to acquire. If you want a "Special Pilot License" (which 'piggybacks' on your JAA license), then no tests will be required. On the other hand, if you want to convert to a regular Australian PPL, then you will have sit a couple of examinations.
The SPL is fine for recreational flying, unless you need instrument or night flying privileges (which cannot be exercised by an SPL holder, regardless of whatever foreign ratings he or she may have). It is a lifetime document.