NHS vs private assessment

Hi, 

I (29F) feel I would like to be assessed as I am fairly sure I am somewhere on the spectrum. I haven't yet made an appointment with my doctor to discuss this as I want to be as prepared as possible (gathering information and reasoning of why my family and I think I am likely to be on the spectrum but also finding out what the process is and NHS vs private options). 

My main questions are:

What is the NHS wait time likely to be after referral (I'm in the West Midlands if that helps)?

Are there any benefits (other than cost and timescale) of the NHS vs private assessment route? I mainly want the assessment for my own peace of mind but if it can help me get any support in any way then I would be open to that as well. 

If I was to go the private route rather than through the NHS, does anyone have any recommendations?  

If anyone has any general information or advice that may be useful please share as I feel slightly overwhelmed with the different information available online!