it can also be from some settings in the startup menu when you boot your computer. When they ask to press F12 or something at the boot to open some settings. I am not a computer wizard, so i do not really know if there is such a setting, but it would be my guess
What exactly is the kind of your old and new batteries? and if your old batterie works fine, why get a new one? Look first if both are compatible. Watch also how you deposit your new batterie inside to see if it is well connected, as a batterie always have a positive and negative, phone batterie always at the same place.
I had the same problem for some times. i tried multiple things like shutting it down and connecting it directly, shutting it down and back up then connecting. But your problem could be also connected directly to Itune. unless there is data that you do not want to lose on it, i would try to reinstal Itune. There is also the Driver that your Iphone use to connect to the computer that could be wrong. As usually, when you connect your iphone, a small driver installation should had started and also ask you if you want to open files or something else. Also, for the best awnser, i would recommend to talk directly with an Apple employee, from their customer services which will point you in the right direction.
The best way to do anything that involves something from Apple, would be to contact the customer service directly and talk to the company and talk directly to someone that is used to repair those problems. i wish i could help more than pointing you to the right direction but it's all i could do. Also, try to head to some Apple sub station, they can also help a bit by sending you to the right people.