Charities refer to other partners all the time. You can tell the person that your data may be shared with an external partner - and record that consent - and then the data is shared.
It's good practice to have a data sharing agreement - but not essential. As long as everyone agrees what the purpose is and what the data can be used for.
I would love to see the Secretary of State quizzed about this . In detail
Or are they going to stop ALL external referrals to any charity?