Once upon a time, the Brits used to laugh in a superior kind of way when they heard Americans pronounce it with 2 syllables. The reason was that Greek was taught in English public/independent schools, and also the top government schools, for much longer than it was in the USA; plus the Continent is right next door to them, so they were used to sounding that final E. However, they did change the I from an EE to an EYE, giving Eye-ree-nee. And the truly awful nickname form of Reenie!
Now it's probable that they've been colonised like the rest of us and, mostly, gone the American route. Here in South Africa people only seem to use Eye-reen.