Getting onto the BV list is not that hard. Move miles from your family, tell your neighbours nothing, then die leaving no clues as to where you are from. The law then requires a proate solicitor to advertise for heirs to come forward in the paper (usually one local to your death) after set period of time no one comes forward, so solicitor pays any creditors and himself then turns the rest over to the treasury who then put it on BV.
Social Services have no part in it. With no family the estate would pay for the burial if enough money exists. Someone will correct me but I believe all those listed on BV are estates over £5000