Shiny new iPhone 5S looks pretty good on the face of it. Once home I connected it to the home wifi and the restore began again.
After a few restarts it finally ended at 0020. Good job I hadn't expected to sit in the shop until it was done.
Most of the apps restored themselves but not of the logins and some of them failed to work properly and needed deleting and re-downloading.
So far. A week later an no problems with it turning itself off.