There is a tree at ancestry.com that shows Beatrice AZ. Williamson born 1920 (just the year) no full parents names but shows her married to Patrick Cyril Weldon.(and shows he mother as maiden name THEW) It shows both born England, both died South Africa.
So with this info I looked fora passenger list at Find my Past and there is one which seems to match pretty well.
Feb. 15, 1922 London to Cape, South Africa Ship: GARTH CASTLE
WILLIAMSON Mr. W born about 1882 Mrs. S born about 1887 Miss B.A. born about 1921 Miss E.I. born about 1920 Master W. born about 1917
for some reason I am not able to access the image for them though - maybe someone else will have more luck
If this is right family I assume Miss E.I. would be Esther Patricia with perhaps wrong initial ??
I took a look at births and see that whoever has that tree took the THEW name from a birth record but there is also a birth record for girl with same name Beatrice A. Williamson, born 1921 Tynemouth, Cumberland with mother maiden name GUY which fits with what you have.
I looked for the other children - William and Esther but did not find them as WILLIAMSON - there is a William F. GUY born 1918 Tynemouth, Cumberland mother's maiden name GUY
I did not find Esther Patricia at all
I also did not find a marriage for Sarah Guy and William Williamson
I just found another passenger list at ancestry.com
this one, if same family leads me to believe that the 1922 sailing I found was not their first trip to South Africa
Here they are in 1921 entering UK
UK Incoming Passenger Lists, 1878-1960 about Sarah Williamson Name: Sarah Williamson Birth Date: abt 1887 Age: 34 Port of Departure: Durban, South Africa Arrival Date: 30 May 1921 Port of Arrival: Southampton, England Ports of Voyage: East London [Port Elizabeth] [Cape Town] Ship Name: Walmer Castle Search Ship Database: View the 'Walmer Castle' in the 'Passenger Ships and Images' database Shipping Line: Union-Castle Mail Steamship Company Ltd Official Number: 114839
WILLIAMSON William, miner, age 40
Sarah, wife, age 34,
Esther P., age 1
all show permanent residence as South Africa Beatrcie is not with them.
Also at ancestry.com I found that 1922 passenger list for them going to South Africa - shown above, and was able to see the image which shows
their adddress in UK:17 Ellesmere Road, Benwell, Newcastle and the Miss E.I. does look like E.P. in image (in the ancestry index she is shown as E.J. so I guess the different indexers read it differently)