Hi jelladlra and all,
I am a very early iPhone6 64GB onwer who experienced exactly the same problem than yours, few days after (but not right after) the iOS 8.3 update.
I experienced this problem few days before your post. As of now there are thousands of posts on the Internet about issues connecting to iTunes & App Store from an iPhone/iPad, but I was able to find less than 10 posts all over the Internet that might eventually correspond to my problem, and one post, Jelladlra's, that perfectly matches.
Logging to iCloud from my iPhone works perfectly with all of my 3 Apple ID's.
Logging to iTunes Store from my MacBook works perfectly with all of my 3 Apple ID's.
However, absolutely no way to login to the iTunes & App Store from my iPhone with any of my 3 Apple ID's. Absolutely nothing happens after login, besides that after pressing the "Sign In" blue-text-on-white-background-button, the blue text becomes gray then blue again.
I tried everything as Jelladlra, and anything else (I think) that can be done, including restoring my iPhone from a backup from another (working) iPhone. I also did the same thing the other way round! All my attemps failed but I could at least confirm the problem was my iPhone, and nothing else.
As a last resort I called the Apple Support. Sparing the details, the final answer was "your iPhone needs to be replaced; here are the options". I wanted to know more and I insisted to know the root cause and I was told what happened to my iPhone is "an electronic failure associated with the iOS update process" but that was as much as I could get as information.
Today I received a brand new iPhone6, just restored from backup, problem gone.
I am sorry Jelladlra for the bad news and I am interested to know more if anyone has anything to share.
- Delphi8000
The following is for Geeks only. The electronics failure answer is a difficult answer for me to swallow, but not impossible. After some Googling I found out rumors reporting that the first iPhone6 flash memories are TLC NAND type and have some reliability issues, whereas the more recent memories are MLC NAND type, faster and way more reliable. This is of course pure speculation but it might eventually explain:
http://appleinsider.com/articles/14/11/07/rumor-faulty-flash-memory-in-select-ip hone-6-6-plus-units-prompts-apple-to-mak…
http://www.eteknix.com/apple-disabled-tlc-memory/
Following app allows to know if you have TLC or MLC memory but unfortunately it does not work for iOS 8.3 at the time I'm writing this:
http://www.pgyer.com/tlccheck
If you jailbrake you can connect as root through ssh and get to know your memory type:
http://www.redmondpie.com/how-to-check-if-iphone-6-6-plus-has-mlc-or-tlc-nand-fl ash-memory-video/
Otherwise you need to open up your iPhone, look at the memory chip, Google the writing on it and get the memory datasheet.
Please let me know if anyone knows how to know the iPhone memory type without jailbreaking or opening the iPhone !
Cheers!