Do you think that it would be more effective if I would speak English, my partner Korean and the nanny Italian?
My recommendation is to speak to the child in the (most) native language of the parent-speaker when approaching OPOL (also see my motivation here). Secondly make sure the child is sufficiently exposed to the language. Annika Bourgogne recommends a minimum 25 hours per week of human interaction/dialogue in each language the child is learning (this is in her book "Be Bilingual", I am quoting the minimum hours by heart). I would take these two criteria in mind when deciding to speak English to your child: do you speak English fluently and would you expose the child sufficient in English? If not, I would recommend you benefit from the Italian speaking nanny and have her reinforce Italian. Together you would make sure the child is sufficiently exposed.
The basis of my recommendation:
Our family situation is somewhat similar: when our first child was born we stuck to OPOL, which both were minority languages. Together my wife and I speak English, which the majority language where we live. At 2.5 years our nanny joined who speaks English with our children. At the time our son was already fluent in both our languages, and within 2 weeks or so after the nanny joined he also started using English. It was quite clear that he had learned English passively from listening to his parents. He did not speak it fluently at the beginning but now a few years later he does as it is the majority language where we live thus he is well exposed.
When our second child was born our family was tri-lingual and our first born continued to manage this well. Our second born had to deal with three languages from the start and managed. At 3 years our second child has a very strong preference to speak the majority language (English), but does understand all three languages well.
Which also comes to my last recommendation: when you are dealing with so many languages (I counted 4 in your situation) you are likely to see a big variance in results among children. Some children will make it look like a breeze (like our first born) and others will struggle (like our second born), make sure you are prepared for the last category and have plenty of exposure to the languages that will be most important to the child. For example, our children have formal education in both the minority languages in addition to their usual education in the majority language. I do wonder how important French will be to your child, as in your question you don't indicate how and when you will expose the child to French.