If you have been working for 5 months on the new contract, then you seem to have accepted it. However, is the "new" contract just an amendment to the earlier and and did it say anything about notice?
You r original contract was a zero hours contract - so you only worked when there was work? What hours do you work now, 10 or is it more?
It seems to me that you are still bound by four weeks notice. Having said that, if you did leave without notice, the company could sue you for the money, but few ever do. But, there is always one........