This note was triggered by meeting two US Americans on our walking trip through Northumberland. They had been cycling, but one of them had an accident and had injured her knee. She refused to go to the local National Health Service hospital to have the knee examined and skyped her Dr. in the USA to have a long distance investigation. The couple had several weeks more vacation to "enjoy". We enquired and were told treatment would then follow in the USA, but the lady was in pain. Whether the two had something against socialised medicine or whether they had no vacation medical insurance so were worried about the cost of treatment we didn't enquire.
We have known cases where visitors from outside of Europe have come to Germany without medical insurance and it can be quite expensive. Treatment in Britain is cheaper, but… Cyclists visiting Britain from outside the EU, however should take out a vacation medical insurance. If anything happens it is cheaper than being without an insurance. British public hospitals charge non residents which wasn't formerly the case, but these days the hospitals want to see an European Health Insurance Card (EHIC) or a credit card. Just don't ask about dentists, who are largely private in the UK. An ankle or knee injury could well make a serious hole in £500. Short term medical insurance cover costs less than this.
Visitors to Britain from countries either in or associated with the European Union can receive an EHIC which gives the holder medical treatment in the various countries of the EU under the same terms as an inhabitant of the country they are in. At the moment this applies to UK citizens as long as the country is in the EU. What will happen after BREXIT is not clear.