I worked in Cyprus in 2004 and regularly visited Nissi Beach. I found there to be no seaweed problems until later on in the season. I really don't know why this was - there had certainly not been any storms (or bad weather of any kind). I would say I started noticing it maybe mid august/early september time and it was there everytime I went after that until end of oct. It was not on the beach at all, but was present in the sea. It was quite thin and slimy, but not enough to put me off going back!! As it is a bay, it just seemed to be there in clumps and not really move anywhere! We found it quite bizarre to be honest.
I have to say I liked Fig Tree Bay in Protaras better, but that is just personal preference. My favourite in the area though, is Konnos Bay - a small bay unknown by most tourists to the area, tends to be mainly locals there. It's at Cape Greco which is between Protaras and Ayia Napa.