Hi KA
Despite having (I think) found Caroline in 1871 and 1881, it didn't add much to the picture!
This is a real toughie and unless she's hiding under a hideous mistranscription somewhere its hard to know where to take this forward. There were only 43 Carolines showing as born Gateshead / living Gateshead in 1891, and of these none was even a close age match to your Caroline (only 3 were over 19!)
Was James a jobbing hairdresser or did he have his own shop? If they were reasonably well-to-do, could he and his family have gone abroad for a spell for some purer air?
There's no obvious sign of any of the children in 1891 either.
You say you have found the family, scattered, in 1901. Does this include Caroline?
Having spent an hour or two puzzling this one, and having consumed more than my usual quota of red wine, I think I have a little appreciation of frustrations you must have faced with this family!
It might be worth collecting all you have got into one posting on a new thread, and hope that the 'big guns' on Rootschat can crack it! I don't feel up to the task I'm afraid!
It might be worth examining the later material you've got again. With such a plethora of Atkinsons in the area its a tricky one.
Where does James show up as a hairdresser? What is your source for him having TB? Where does the 'probable' Robert come into things?
Maybe James is hiding out there, but he's not making himself obvious yet!
Good luck and best wishes
Paul