Lisa, you don't need a referral to see a private menopause specialist. If you can afford it, I'd highly recommend it. You can ask them to write to your GP and tell your GP what to prescribe for you so that your meds and future care can revert to the NHS and you may just need to pay once...
I see Dr Olivia Jones at Newson Health and would highly recommend her. It's about £230 for the consult. Once you are stable on HRT, you need to be seen once a year to stay in their system but you don't need to get your meds from them. It is about £175 to see a nurse or pharmacist there who can also prescribe HRT but if your case is a bit complicated, it's best to see a doctor. (I tried to save money once by switching to a nurse but because I'm a bit complicated, she referred me back to a doctor there. They didn't charge me again but I had to do 2 consults....)