OK, so I have found a solution that worked for me ...
Just to set the scene - i have three AppleIDs i use, for a number of different reasons ... it's madness, but when I started with Apple i got my pants in a mess and didn't really know what I was doing, but I am living with it quite OK.
I have m.p@mac.com for iCloud and all things associated
I had iphone@mp.co.uk for personal iTunes / Apps etc
I have applestore@mp.co.uk for my photography related business apps
(all email addresses abbreviated for demo purposes)
All of a sudden, some weeks ago (after the 9.3 update - i think it was this one), my iphone@mp.co.uk username & password combo were not recognised on my iPhone 6 Plus. No idea why. It has continued for some time and had gotten to the point today where I had 17 apps to update, but no way of doing so ...
If i used the same account to login on the Apple Website, it worked.
If I used the same account to login via my iPad, it worked.
Frustrating, as many of you have experienced.
I tried all the previous suggestions ... reboot / shutting off iMessage / logging out of everything / Reset All Settings (painful) / Changing password .. nothing worked.
So then this afternoon I came across a brainwave ... and I changed the email address associated with the Apple ID. I have been meaning to create me a new gmail account anyway, so did that first, checked mail was flowing, logged into this URL 'https://appleid.apple.com/account/manage' and changed the email address of my Apple ID to the new one I just created.
Attempted login with my iPhone 6 ... BLAMMO!!! First time, in and all apps updated.
Of course, that's a real PITA as I have iphone@mp.co.uk like EVERYWHERE, but a small price to pay compared to the frustration i have been feeling these last few days. (At least the new AppleTV has Siri dictation to make this easier 😉 )
So, that's what I did - i really hope it helps everybody who is still suffering with the problem. Basically, change the email address associated with the AppleID, and see what happens.
Good Luck !!