I have found that a John, Fred 'Archbold' and Florence Frater were admitted to the Newcastle Workhouse. The other children you have mentioned are not there. This is the information from the Creed Registers - I will give the references:
HO.NGH/88/1: 1 Oct 1912 - 25 Oct 13
Admitted on 10.1.13
Frater, John, 1902
Frater, Fred Archbold , 1904
Frater, Florence 1906
The years are under the heading of 'From whence admitted' but I think these may be their dates of birth. They were all E.C. religion.
Name of Informant:
Grandfather. John Cowe, Stations Cottage, Berwick on Tweed
It looks like Fred was discharged on 3.2.13.
In the next register HO.NGH/88/2: 25 Oct 1913 - 25 Oct 1916 ...
John and Florence listed as 'Orphan' in John's line and are both still in the workhouse. Grandfather & same address as above listed again as their relative. John and Florence were discharged on 28.3.14
There is a very small entry besides the entries for John and Florence but does not give a date which reads:
'1 hr Fred A in house'
Unfortunately, the Creed registers give no indication where the children went. As Fred is only down for having been in for one hour I wonder if he may have been immediately sent or taken elsewhere?
Possibly to the Wellesley? I wonder what Christine might find? I can tell you that he did not go to the Axwell Park school - I have checked and could not find him there.
This is the Elswick Cemetery burial entry for Eleanor:
Entry 44266
Eleanor Frater
Female
Age 42
53 Tulloch Street
Buried January 4 1913
Ceremony by Thos. Mitchell
Buried Section Y Grave No 740
Removed from St Stephen's parish
Newcastle Evening Chronicle death notice - Friday, January 3 1913:
'FRATER.- At 53, Tulloch Street, December 31, Eleanor, widow of the late Archibold Frater. To be interred Elswick Cemetery, on Saturday, 2.30 prompt. Friends please accept this the only intimation.'
This is a map of the Elswick Cemetery. Click on thumbnail image to enlarge:
http://www.margaret-hall-genealogy.com/page8.htm