I am aged sixteen or seventeen and I want to start puberty blockers on the NHS. Can I consent on my own behalf?

Possibly. In the UK, people aged sixteen and over are presumed to be able to consent to medical treatment on their own behalf.
However, the judgement of Bell v Tavistock ruled that in some cases clinicians may find it appropriate to involve the court in the referral of a sixteen or seventeen year old for puberty blockers. If this is the case, you may need to apply for a court order to continue with treatment.