Preparing tax returns for a state-to-state move like yours is very complicated. I'll give you my thoughts on it, but I'm certainly not an expert on such situations. If it gets too confusing, you might want to go to a tax professional for this year. (But even that's not a guarantee. See my personal story at the end of this post.)
Technically any money you earn for working, including a bonus, is taxable by the state or locality where you did the work. That would mean allocating the bonus partly to NY (including NYC) and partly to NJ. But to be practical, I would just report it the way it's shown on your W-2s. That approach is the least likely to raise any questions.
Your description of your W-2s is a little confusing, but if I understand correctly, in effect the employer reported the entire bonus as New York income (but not NYC). I would just enter the amounts as they appear on the W-2s, but enter it as if it were all on one W-2, with one line of state information for NY and NYC, and a second line of state information for NJ. Don't enter three separate W-2s in TurboTax.
You didn't say exactly when you moved from New York to New Jersey. Unless you moved on January 1 you are going to have to file a part-year resident return for NY in any case, even if you only lived in NY for a small part of the year. NYC tax is included in the NY state return. Your NJ return will also be part-year resident.
When you do your NY return you will have to specify how much you actually earned in New York State. The amount you earned in NY should be your base pay allocated proportionally based on how long you worked in each state, plus the bonus. When it asks how much you earned in New York City, just leave the total NYC wages from the W-2. That's technically not correct, but that's the way the employer reported it, and I don't see any way to add the bonus to your NYC income if it's not included in box 18. So to answer your original question, if you do it this way you will not be paying NYC tax on the bonus.
I don't think you will get any credit on your NJ return for tax paid to another jurisdiction because, if I understand correctly, you did not have any NY income while you were a NJ resident. Therefore you are not paying NJ tax on any of your NY income. Your NY income is taxed only by NY and your NJ income is taxed only by NJ. There is no double taxation, so there is nothing to give a credit for.
(As an aside, here's my personal experience with a similar situation. Many years ago when I moved from New York City to New Jersey I had my tax returns done by a mid-sized accounting firm, i.e. not just an individual accountant. New York disagreed with the way my income was allocated, and wanted me to pay a small amount of additional tax. The accountant I dealt with at the firm said the rules are unclear and I should just pay the additional tax. Where the rules about how to do the allocation were unclear, they did it in the way that favored me, but the state disagreed. So it's tricky, even for the experts.)
