I've also just checked with a Facebook friend who is an Australian nurse who has been working in the UK for just over four years. She had an exemption from the English test, because she was born, lived and studied in an English-speaking country.
I know that when I was employing teachers from abroad, Australians, NZers and South Africans etc didn't have to do the IELTS test, whereas teachers from Russia and other countries outside the EU did. All teachers had to do the skills tests in English and Maths. It was a bit of a minefield. Even if a Russian (or some other EU countries) had a degree in English, he/she would still have to do GCSE English, Maths and Science, because it's a requirement for all teachers employed in England and Wales.