There was no government funding for 'reformatories', but there were charities for boys and girls like the ‘Temporary Refuge’ attached to the London Refuge for the Destitute (more info here on London Lives).
Another early charity was the Philanthropic Society (1788): it aimed to reform criminal boys and girls, and to provide religious and moral education for the children of convicts. The Society, initially based at Hackney (it moved to Southwark 4 years later), took children between nine and twelve years old and gave them industrial training. In 1845 the Society stopped taking in convicts’ children, and female offenders, and concentrated on rehabilitating young male offenders. Four years later, the Society moved from Southwark to Redhill, where it founded a farm so that boys could learn agricultural skills to prepare them for emigration. The boys cultivated the land and cared for cows, horses, sheep and pigs.
The good work done by reformatories persuaded the government to confer official status on institutions approved by the secretary of state. Under the Juvenile Offenders Acts of 1853 and 1854, magistrates could send convicted children over ten years old to certified reformatories for two to five years instead of imposing a long prison sentence. You can find out more about reformatories and their records in Tracing Your Ancestors' Childhood.
Image from author's collection: The Philanthropic Society’s farm at Redhill. Illustrated London News, 14 June 1851.