It is important to be faithful to the way you see information recorded, and that you are giving a forum to use.
The children are registered at birth, WA, as COLE, not COLES.
Rose and Fred, census 1901, are COLE not COLES.
If these children were separated early in life it is possible there would be a variation in how they spell their name.
William Henry in the Coventry and Warwickshire Hospital........can you locate that more clearly please...I am not seeing it. How is he named....age, birthplace.
Frederick George's military record....how is mother named.....Mrs COLES. Mrs Ellen COLES, Mrs E COLES.....what......exactly as recorded please.
What was Frederick George's address at enlistment.....same as for mother?
Is there an address for him when he is discharged?
Who are the witnesses on his marriage certificate?
They went to Australia, 1886..........so, no children for five years, then three children in seven years?
"...I found Frederick and Ellen travelling to Australia in 1886 ..."
How do you see them named on the "Elderslie".......exactly as you see the names please.
You need to see the 1896 birth certificate.