A word of caution about "speaking Cockney". Italy is full of regional dialects and pronunciations. They are very interesting for foreigners and I understand some of them, but I would never dream, even after 30 years in Italy, of trying to speak these dialects except for a moment, just for fun.
Unless you are completely fluent in English and have an excellent ear, you will sound ridiculous.
It's different for people in immigrant communities who have absorbed the language they hear around them, in an almost unconscious manner. They speak Cockney mixed with their own accent, but in a more spontaneous way, which may be quaint but not ridiculous.
I agree with Nona that you should avoid rhyming slang!