There are a few Thomas White deaths in 1870-72, various ages, none born London, too hard to tell if one of them is him.
mum mum
An online tree says that he was the one who died 1872, parents William and Norah [Honorah] McCarthy Born Surrey 23 years old from the index.
From that tree, henry William White born 1871, Redfern NSW reg no 4143.
Now we are looking for an arrival 1870ish for Thomas and Elizabeth White,