Same in Denmark. I'm not a Danish citizen, but since I'm legally resident in the country (work/residence visa), I'm in the system. In fact since everyone is in the system, the yellow health-coverage card is de-facto used as your ID card for some purposes (there's no national ID card).

Anyone visiting the country but not resident here is also entitled to emergency care, e.g. if you fall off your bike as a tourist and have to go to the hospital, there is no charge. Same if you have a heart attack while visiting.

If I get a travel visa to England, I can get free cancer treatment?

If you are entitled to live and work in England, you are entitled to free healthcare. Since the subject is about running a startup in London, you'd have to have those entitlements. This isn't a discussion about visiting England as a tourist.

It depends!

Am I entitled to NHS treatment when I visit England? http://www.nhs.uk/chq/Pages/1086.aspx

Am I entitled to NHS treatment when I move to England? http://www.nhs.uk/chq/pages/1087.aspx

you need right of residence to register for a National Insurance number, but after that you're good to go