Hi unicornpoop
You don't need to see a private doctor to get Estradot patches (and unless you have a Mirena coil you need a progestogen to protect the womb) but it is hit and miss whether you will also get testosterone on NHS as there is no product available that is licensed for women. It is important to make sure you replace oestrogen at an adequate level before introducing testosterone so the two hormones are balanced.
Your reduced libido may well point to low testosterone - unless you are taking meds such as anti-depressants for example which also lower libido. The increased hairiness on face shows that your oestrogen level has dropped considerably and more than your testosterone ( this is particularly noticeably around menopause) - so the ratio of T to O increases - leading to some unwanted androgenic characteristics such as facial hair, fat around the middle rather than hips. Provided you replace oestrogen adequately first for a few months and see how you feel, then adding a small amount of T should be beneficial and not lead to side effects ( except hair growth at the application site possibly). If you look in the Private Lives section there is a long-running thread I started called "Testosterone Here we go" where some of this is discussed. It goes off topic for a bit but then comes back.
Re facial hair - I've never had much unwanted facial hair - just a few dark hairs on upper lip and one hair between lower lip and chin which I have to pull out every now and again (!) - but these were coming before the T anyway. However I never was a hiary person ( in the wrong place) and had lots of it where it should be ie head, underarms etc, and hardly any on legs (I do have hairy thighs now - but easy to shave these off!).
You will probably also need something like Vagifem or an estriol cream for the vaginal area, as well as the HRT - all available on NHS. If you look at the Treatments tab on the menu it gives all the vaginal preparations.
It would be a good idea to have a basic blood tests though - not for menopause but to check if there other reasons for your hair loss and sleep problems - eg vitamin/mineral deficiencies, thyroid problems etc so do ask for these when you ask about HRT.
Hope this helps
Hurdity x