This does not address your enquiry about birth details for Elizabeth but might add generally to what you have, enlistment in Australian Navy 1914 for Thomas.
NAA Service (Petty Officers and Men)
Thomas John ROGAN b. 20 Mar 1896 Liverpool, England
20 Apr 1914 engagement for 5 years.
Next of Kin wife, Elizabeth Prince William St, off Hill St Liverpool, England.
Later addresses are -
62 Wells St Annandale 19 May 1919
222 Hereford St Forest Lodge Sydney
144 Morgan St Merewether, Newcastle 7 Feb 1920.