My NHS number changed when I changed my name. I did not change anything else doctor, address etc.
The new rules that are supposed to have been adopted means that your GP can now refer you direct. Whether Charing Cross will abide by them is another question.
I have used Gendercare and they were excellent. The staff actually work at Charing Cross. Obviously they are expensive, as is all private care. I am waiting to hear from Charing Cross. I sent back my forms in early July but have heard nothing yet. I put two first class stamps on but now I am worrying that they may not have received them and I will be back at the start again. By the way it took the psychiatrist/doctor nearly six months to send the referral. I was constantly assured it had been done but it only happened when I sat down in front of my doctor and watched him do it.