Looks like the 2 children, Thomas and Daisy were orphaned by 1901 as they both appear as Heally at St Marylebone schools at Southall.
The admission/discharge register for the school at Southall shows that the children Thomas, Cissy and Daisy were admitted in January 1900.
Both parents are recorded as dead, the father having died in the Infirmary. Their grandmother was
Eliza PAUL, 21 Wakefield Street, Grays Inn Road.
Thomas HEALLY, aged 12, date of birth given as 17 May 1887, was admitted on 12 Jan 1900. He was discharged on 20 September 1902 to the 5th Northumberland Fusiliers, but returned to the school 2 days later on 22 Sep 1902 'from enlistment at St George's Barracks'. Then sent on 29 Jan 1903 to Mr D. R. Evans, Ellesmere College, Salop.
Cissy HEALLY, aged 8, was admitted on 20 Jan 1900, no date of birth. She was discharged on 16 Feb 1907 to Mrs Wickers, 'Bolingbroke', Uxbridge Road, Ealing (perhaps into domestic service?).
Daisy HEALLY, aged 10, was also admitted on 20 Jan 1900. She was discharged on 13 July 1904 to 'Notting Hill Infirmary' (the Marylebone Union Infirmary).
There will probably be more school admissions/discharges for these children, and you may want to look for yourself (on Ancestry). The path for this particular register is: London, England, Poor Law and Board of Guardian Records, 1430-1930 > Westminster > St Marylebone > Admission and Discharge > Admission and Discharge Registers, Southall School, 1897-1908 > image 117.
There are several other admission/discharge registers for the workhouse/infirmary and school that are online and might also be worth a look, if you have access.
For information about the children's emigration, you could have the Board of Guardians minute books searched at LMA (these are not online).
