The two gentlemen in question were my grandmother's half-brothers. According to oral family history, both had emigrated from the UK to the USA. My father said they may have entered the States via Canada; to the best of his knowledge, one or both settled in the Boston area. Ernest Charles may have gone by his middle name -- my uncle once referred to him as his 'Uncle Charlie'. Ernest Charles was born in Stepney; he was listed on the 1901 census, living at his father's home (occupation, gas company labourer). His presence in the UK has not been verified as yet on the 1911 census. Walter Henry was born in West Ham -- as of the 1911 census, he was living at home (occupation, Clerk). Their father's name was Frederick Samuel Parsons (1850-1926); their mother was Sophy Ann Long (1847-1891).
Sincere thanks in advance for any info or leads you can share.
edit: thank you for the first answer -- I should have specified: I'm hoping for leads on the boys' immigration -- whether they settled in the US (or Canada), whether they returned to the UK, etc. After 1911, they seem to have vanished!