Yes, you are correct.
Certain practices will have an allocated budget. Shared care is generally used as the go-between, from the private treatment to the GP. Usually private will assess and update your doctor every medication review, and your GP will write the prescription for the medication saving you the cost. However, certain practices simply cannot do this. They need total control to observe and prescribe. The biggest issue is Doctors are not educated enough on ADHD, and can’t prescribe drugs which are considered pretty powerful. The NHS has certain procedures to follow. However, not every practice will decline. Some will happily accept shared care. But many of them will not. It’s also a liability thing, which they must make sure they’re covered.
You can be assessed by the best and most well-known specialist, and at the end of the day, if the practice does not want to be liable for prescribing these medications, they simply won’t accept it. However, that doesn’t mean it’s the end of the world.
Once you go private and you have been diagnosed, it makes the job a lot easier for the NHS. You can bring all your documentation to the NHS equivalent specialist, and it should theoretically expedite the process. They shouldn’t ignore a private assessors diagnosis, simply because it’s private. They also technically have a lawsuit on their hands for discrimination. However, that’s messy, and not advised. But it’s still technically possible.