The ash bundles contain ash gathered from a human cremation site wrapped in animal skin, and were thought to have been used as amulets to ward off illness.
They were acquired in about 1838 by George Augustus Robinson, a Christian missionary and conciliator for conflicts between Aborigines and early settlers.
He was sent to find the dwindling number of Aborigines in Tasmania in the 1830s and take them to safety on Flinders Island.
The bundles passed from the Royal College of Surgeons to the British Museum in 1882.
The decision to return the remains to the Tasmanian Aboriginal Centre had come as a result of the British Human Tissue Act of 2005, which for the first time enabled museums to repatriate remains.
Previously they were unable to consider requests by Aboriginal campaigners.
The British Museum said in a statement "After taking independent expert advice on the matter, and according to their published policy, the Trustees came to the view that the cultural and religious importance of the cremation ash bundles to the Tasmanian Aboriginal community outweighed any other public benefit that would have flowed from their retention in the collection."