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

Unfortunately this is unlikely. The UK has a legal provision called Gillick competence, which means that children can consent to treatment provided they fully understand the consequences of the treatment. However, the judgement of Bell v Tavistock has led the NHS not to accept informed consent from young people in regards to transgender healthcare, even where Gillick competence might otherwise apply.
If you are supported by your parents (or people with parental responsibility), then they may be able to consent on your behalf. An independent panel will still need to review your case when you are referred for blockers. 
If you are not supported by your parents, or the independent panel declines your referral, you will need to apply for a court order to continue with treatment.