How to change nickname when writing reviews in App Store?

Nope; if you search the Apple FAQs and other discussion threads, your login and posting ID are tied together. The best solution is to have one ID for purchases, and one for comments. You just have to login using the old account if that is the nickname you want to be displayed.

I don't like the ID that my Nickname is my login name, and I have given people half of my login credentials already.

Apple, other forums allow for a display name different from login name (hint!)

Are you still able to log into your old account? You should be able to change the nickname in the old account to something else, and then iTunes should let you add the old nickname to the new account. (I was able to do this recently, but I had only added the nicknames within the previous few days).

If you didn't already sort it--

in iTunes, go to Store | View My Account. Make sure to enter the email for your old account.

The main Apple Account Information screen has an Edit Nickname button next to your nickname. You should be able to change that nickname in the old account.

Then log out of the Store, log back in with the email for your new account, and see if it will let you use the old nickname.

It might be necessary to delay while the computers figure out the old name is free, if it doesn't work immediately.