When an NHS organisation is delivering a service which is commissioned and funded by a non-NHS organisation (e.g. a local authority), the design of communications is the commissioner’s decision.
There are different approaches that the commissioner may choose to take, depending on what is felt to be in the best interests of people accessing the service.
They could choose to lead with the commissioner’s identity/visual style or with the NHS identity to reflect the organisation that is providing the service.
Please see the visual example for local authority commissioned and funded services.