After much searching, this appears to the ship William went to Australia on, to find work and found a home for his family:
 

MERCHANT SHIPPING ACT, 1906, and ALIENS ACT, 1905

OUT-GOING PASSENGERS

Returns of Passengers leaving the United Kingdom in ships bound for places out of Europe, and not within the Mediterranean Sea.

NOTE.- All Passengers carried by such ships are to be included, whether proceeding to European or non-European Ports. 1st Class, 2nd Class, and 3rd Class Passengers are to be entered in separate groups. Children born of alien parents in the United Kingdom should be entered in the alien portion of the Return, and the fact that they were so born should be stated in the nationality column.

Ship's Name: "Wakool"
Official Number: 109,996 XNK
Steamship Line: P. & O. Branch Service
Master's Name: Hamilton CALL
Registered Tonnage: 3,147
Aggregate Number of superficial feet in the several compartments set apart for Passengers, other than Cabin Passengers: 5687
Total Number of Statute Adults, exclusive of Master, Crew, and Cabin Passengers, which the Ship can legally carry: 377
Where Bound: Australia via Cape Town

I hereby Certify that the Provisions actually laden on board this Ship are sufficient, according to the requirements of the Merchant Shipping Acts, for 287 Statute Adults for a voyage of 63 days.

Date of Departure: 27th August, 1912
(Signature) Hamilton CALL, Master
Date: 26th August, 1912

Names and Descriptions of BRITISH Passengers Embarked at the Port of LONDON.

Contract Ticket Number: 319
Names of Passengers: William DEER
Class: 3rd
Port at which Passengers have contracted to land: Adelaide
Profession, Occupation or Calling of Passengers: W'houseman
Ages of Passengers: 35
Country of last Permanent Residence: England
Country of Intended Future Permanent Residence: Australia

SUMMARY OF BRITISH AND ALIEN PASSENGERS

British Subjects:
    Accompanied by husband or wife: Males 28 - Females 28
    Not accompanied by husband or wife: Males 145 - Females 52
    Children between 1 and 12 years: Males 17 - Females 20
    Infants: Males 5 - Females 7

TOTAL:
    Accompanied by husband or wife: Males 28 - Females 28
    Not accompanied by husband or wife: Males 145 - Females 52
    Children between 1 and 12 years: Males 17 - Females 20
    Infants: Males 5 - Females 3

Total Number of Adults: Souls 253 - Equal to Statute Adults 253
Total Number of Children between 1 and 12 years: Souls 37 - Equal to Statute Adults 18½
Total Number of Infants: Souls 8 - Equal to Statue Adults --
GRAND TOTAL: Souls 298 - Equal to Statute Adults 271½

Aust. & Cape 298

We hereby Certify that the above is a correct List of the Names and Descriptions of all the Passengers who embarked at the Port of LONDON
Signed [Hamilton CALL] Master
[P. O. GRIFFITHS] Asst. Emigration Officer
Countersigned [John W. DURHAM] Officer of Customs and Excise at LONDON
[Stamped] Collector, Customs & Excise, Custom House, London, EC., Export Branch 16 APR 1913

____________

 
Carrie and Harold followed a year later:

Name of Ship: "Benalla"
Date of Departure: 17/4/1913

Names and Descriptions of BRITISH Passengers Embarked at the Port of LONDON

Contract Ticket Number: 515
Names of Passengers: Mrs. Caroline DEER, Harold do
Class: 3rd
Port at which Passengers have contracted to land: Adelaide
Profession, Occupation or Calling of Passengers: Married
Ages of Passengers: 33, 6
Country of last Permanent Residence: England
Country of Intended Future Permanent Residence: Australia

SUMMARY OF BRITISH AND ALIEN PASSENGERS

British Subjects:
    Accompanied by husband or wife: Males 109 - Females 109
    Not accompanied by husband or wife: Males 340 - Females 224
    Children between 1 and 12 years: Males 112 - Females 124
    Infants: Males 12 - Females 7

Aliens: Non-transmigrants
    Accompanied by husband or wife: Males 1 - Females 1
    Not accompanied by husband or wife: Males 2 - Females 4
    Children between 1 and 12 years: Males 2

TOTAL:
    Accompanied by husband or wife: Males 110 - Females 110
    Not accompanied by husband or wife: Males 342 - Females 228
    Children between 1 and 12 years: Males 114 - Females 124
    Infants: Males 12 - Females 7

Total Number of Adults: Souls 790 - Equal to Statute Adults 790
Total Number of Children between 1 and 12 years: Souls 238 - Equal to Statute Adults 119
Total Number of Infants: Souls 19 - Equal to Statue Adults --
GRAND TOTAL: Souls 1047 - Equal to Statute Adults 909

1047 Aust. & Cape

We hereby Certify that the above is a correct List of the Names and Descriptions of all the Passengers who embarked at the Port of LONDON
Signed [N. C. SIMONDS] Master
[F. M. BATE] Emigration Officer
Countersigned [John W. DURHAM] Officer of Customs and Excise at LONDON
[Stamped] Collector, Customs & Excise, Custom House, London, EC., Export Branch 16 APR 1913