"As set out in the Local Government (Household Charge) Act 2011, liability to pay the charge is determined on the basis of ownership of the property in question on a single day each year. This date, which is called the "liability date", is the 1st of January. The charge must be paid within three months (31st March) of the liability date in order to avoid late payment penalties."
More than likely the buyers' solicitor will want proof that all charges on the property (incl. household charge) are paid up to date at exchange.