Yes, to get the ball rolling, you need to talk to a GP. It can't hurt to have some stuff printed off from NHS web sites, but it is not mandatory. Patient confidentiality applies in all of your dealings with your GP so yes a visit to discuss Gender Dysphoria is equally protected.

Your GP will then refer you to a local Mental Health practitioner (still in confidence) who will screen you for mental health issues that may need to be taken into consideration whilst treating your Gender Dysphoria.

After that, your GP will be able to refer you to the nearest Gender Identity Clinic (in your case Charing Cross) but be prepared to wait quite a while for your first appointment.

In the Charing Cross Gender Identity Clinic FAQ, you can see
Do I have to dress/act a certain way?

No. Everyone is different. Dress and act as you feel most comfortable. This is a safe space and anything that people generally wear in public is fine.
I may have misunderstood your last question, but normally HRT will be prescribed by the GIC only if they feel it will help you to manage your Gender Dysphoria. In many cases this will be through transition.