If the children are primary age they will be fine in the state schools. They will get extra help and will soon pick up the Greek language and make friends
My niece is 7 years old in a state school in Paphos. The english schools were quite expensive (American academy is the cheapest I believe at around £2k a year) and an English teacher said the same think about picking up Greek in no time. This totally didn't happen. It is one and a half years later and she understands pretty much no Greek. Maths is fine as you can't really go wrong with numbers, but everything else she is stuffed. Apparently she comes home every night crying, mainly because she cannot understand anything. My sister asked the teachers if they could translate some of the tests to English to help her out, but they won't do it. Don't blame them either. When one or two kids in the entire school are english speaking only (she also speaks Welsh, but lets not kid ourselves...) then why should they bother? It was similar (but easier because bilingual is the law there) when we grew up in Wales.
When they realised it wasn't working they took her to one of the english schools, but she had lost so much time and because she isn't the greatest at tests, that she basically failed the entry exam. Erm....
Ok, so my sister is to blame a bit for living up a mountain in the middle of nowhere in Paphos, plus moving around a lot so my niece isn't the sharpest knife in the drawer, but unless your kid is pretty clued up, *and* there is at least a smattering of english speaking kids of a similar standard in the same year, then I personally would get them into the English schools. Hang the price. Which is why hopefully my sister stops renting the villa with the pool up on the mountain and comes to Limassol where the rest of us lives (I mean my family and my parents family).
Cheers