Note that it is the solicitor, not the client, who engages the barrister. Assume that the disclosure requirements have been met. When the matter is completed, the barrister renders a bill to the solicitor. The solicitor generally chooses to pay this from the office account. The procedure followed needs to comply with Rule 42(5) and (8) of the Legal Profession General Rules 2015 . In other words, the office account is debited when the solicitor has paid the fee on behalf of the client. The solicitor may then transfer the money out of the trust account after having given or sent the client a request for payment or a written notice of the withdrawal.
For a payment to be made direct from trust to the barrister, a bill would need to be given to the client by the law practice and the procedures for R 42(3) would need to be met.
See also the FAQ in LIV Costs Law Resources.