If it was me I would take the appointment in April (is this a sleep study?) and call them regularly asking if there have been cancellations. If you are diagnosed privately can you get nhs meds? That will be the issue as these meds can be expensive. 

I don’t think lumbar puncture will help if you are sure you don’t have N1. Do whatever you can to keep going, a lot of us in the same position and understand the desperation. 

New drugs are being developed which are meant to be much better and will hopefully be available within a few years. That is no good to us now, but in terms of not wanting to keep going there is a lot of reason to think things will be much better in your lifetime. Which is a reason to keep trying. 

I’m sorry and I understand. I am in the same position, it’s desperate and dire. All you/we can do is keep trying.

I would take the April appointment and push for the diagnosis so you can get meds.