Update - the UK has now left the EU, and all specifics of this answer w.r.t. EHIC cards and EU citizen access to the NHS is likely incorrect. I'm leaving the article since some of the arguments are unchanged, but the argument for health insurance is probably much stronger. (But you still won't be left to bleed to death while somebody tries to check your insurance status).
Provided you are an EU citizen with an EHIC (essential!) , then the UK is one of the better places to be without health insurance for "something really serious". You will get emergency treatment from being carried into a (free) ambulance until you are able to walk (or limp) out of the hospital, and you won't have to pay anything for it. You probably couldn't find better treatment privately, if you needed treatment within minutes to save your life. You certainly won't be left bleeding to death while somebody tries to check your insurance status.
On the other hand, our health service is creaking badly, and non-emergency treatment is effectively rationed by making it slow and cumbersome to obtain. So if you just sprain or maybe-break a wrist or ankle, it will cost you a considerable amount of time (= lost holiday? ) to get it checked out and bandaged up. With health insurance you could probably get it checked out and patched up faster and be back to enjoying yourself to whatever extent you are still able. You might also get better treatment.
Also our health service won't pay for the travel back home if you want to return early or miss your booked flight while in hospital. Neither will it pay for extra days in a hotel. So if you could overstay and be looked after by family or good friends, relying on the NHS alone may be OK. If you are on a limited budget and staying in hotels, then I'd strongly recommend insurance. For a 16-year-old with no pre-existing health issues, it should be pretty cheap. (It wasn't particularly expensive for this 59-year-old with declared medication for hypertension, visiting the EU).
Also, travel insurance should reimburse you for your wasted bookings, should something cause you to have to cancel your trip. Your illness, or that of a relative you were going to stay with or who you suddenly need to care for back home. (Some policies have more exclusions than others on this front, read carefully!)