Just in case - two of these children from the 1901/1911 Census immigrated to New Zealand
Nora Margaret LAVERY & Gerald Leo Gregory LAVERY.
Nora Lavery b: abt: 1889 sailed from London to New Zealand Wellington in 1913
Gerald Lavery b: abt: 1892 sailed from Southampton to New Zealand Wellington in 1925
New Zealand BMD
Reg# 1915/14622 Nora Margaret LAVERY married Charles Alfred SMITH
Reg# 1952/23506 LAVERY, Gerald Leo Gregory age: 61 years
In a New Zealand paper there is a picture of Gerald Lavery in 1933. If you think this might be your family let me know and I will send more information.