Tom's Modelworks and bigjimslade are right. I just took a more careful look at the Sumrall book, which sorts out the following details.
The New Jersey was commissioned on May 23, 1943 with the open bridge arrangement, which lasted through the first portion of her post-commissioning trials. The enclosed, round-fronted enclosure was added by October 13 of that year (the date on a couple of the photos in the book). She headed for the Pacific early in January, 1944. The round, enclosed bridge lasted through her first operational deployment, which included the Caroline, Marshall, and Marianas Islands campaigns, the Battle of Leyte Gulf, the December, 1944 typhoon, and the pre-invasion bombardment of Okinawa in March, 1945. She left the war zone on April 16, 1945, and headed for Puget Sound, where she underwent her refit in May and June (and got the Missouri-like square bridge front). She then returned to the Pacific, taking part in one more shore bombardment, at Wake Island, before the war ended.
My earlier post was in error in saying that the NJ was in attendance at the Tokyo Bay surrender ceremony. In fact she got to Tokyo Bay on September 16 - two weeks later. Sorry about that; my fault.
In addition to the question of the bridge configurations, there were changes in the anti-aircraft armament of all the Iowa-class battleships that might be of interest to model builders who are really picky about such things. Sumrall's book (which I trust pretty thoroughly) says the Missouri and the New Jersey originally carried forty-nine 20mm single gun mounts apiece. He says there's some documentary evidence that each of them also had eight 20mm twin mounts as of April, 1945. Sumrall admits that the evidence on this point is murky, though; he hasn't found any photos to support it, and that date doesn't make much sense in terms of the ships' visits to dockyard facilities. Sumrall also says the Missouri's complement of single 20mm mounts had been reduced to forty-three as of September, 1945. The Tamiya 1/700 kit (I don't know about the 1/350 one) has forty-nine single 20mm mounts and no twins. If I were building either the Missouri or the New Jersey in wartime configuration I'd leave that arrangement as-is.
Then, of course, there's the confusing question of radar screens, which matter to some modelers and not others. For those who do care about them, I strongly recommend the Gold Medal Models photo-etched set. In addition to amazing reproductions of all the relevant radar screens, it has a fine instruction sheet summarizing when the ships got the different sets. If you're interested, the Tamiya 1/700 kit has the round, SK-2 screen on the foremast, which is correct for the Missouri in 1945 but, according to GMM and the Sumrall book, not for the New Jersey until late 1945. Prior to that time the NJ carried the older, rectangular SK aerial. The good news is that, if you're using the the GMM parts set, the SK is easier to build. The framework on that SK-2 is pretty scary, at least to these 53-year-old eyeballs.
The bottom line remains what therriman said last week. If you build the Tamiya Missouri kit out of the box, with the "Measure 22 Graded" color scheme illustrated in the instructions, and change the hull numbers, you'll (probably) have an accurate model of the New Jersey - as she appeared in late 1945. If you want to represent her during her lengthy Pacific deployment in 1944, you'll need to modify the bridge front, change the radar screen on the forward superstructure, and paint her in the "Measure 21" color scheme (navy blue on all vertical surfaces, deck blue on all horizontal surfaces). Otherwise, if I'm reading the references right, the details in the kit are right for both ships in both periods.
Fascinating stuff. Welcome to the wonderful world of ship modeling. As you can see, it's populated by strange people who sometimes get obsessed with utter trivia and, in particularly obnoxious cases, spend their time during their summer vacations sitting at computers writing interminable forum posts. In most cases, fortunately, such people are relatively harmless.