Has the private psychiatrist said why? It may be because it is usually considered bad practice for a person to be under the care of more than one practitioner, which does make sense.
The issue that you may face is 'being within Camhs' can mean so many different things and your son unfortunately may not be seen by any professionals.
My son also attended A&E at 14 due to a MH crisis. Camhs had previously refused to be involved as he wasn't 'unwell enough' despite having suicidal ideation and psychosis.
After the A&E visit he was cared for by the Camhs crisis team who visited us at home twice a week for 8 weeks.
They were great, but their focus was supporting him through the tough stuff and keeping him safe rather than any type of diagnosis, meds, talking therapy etc..
We were initially assured he would see a psychiatrist and a psychosis team. Unfortunately, they decided after a few weeks that he didn't meet the criteria for either (which is appalling as he was very poorly), and even if he did the wait time was over 9 months.
His crisis caseworker suggested a private physiatrist who was great and diagnosed adhd driven anxiety. This was a huge turning point for DS.
OP please don't let the psychiatrist assume that another psychiatrist will see your son anytime soon, or anytime at all via Camhs.
Do also bear in mind that a psychiatrist does tend to take a medicine based approach to MH issues, or disabilities like adhd . This most definitely has a place, and DS has found adhd meds transformational. However, talking type therapy can be valuable and is usually not part of a psychiatrists tool kit.
Adhd specific counselling and coaching has also been incredibly helpful for DS. This must be done by someone with lived experience of adhd IMO. This isn't delivered by Camhs in our area unfortunately, but they have been very helpful wrt meds.