If you have health coverage as part of a national plan in the EU, or have a health insurance covering your basic health cost in the EU, you can get a card that shows this and that will cover your emergency health care while traveling in the EU.
Living outside Italy it might well be that you are not covered under the national health system.
If your national or private health insurance does not cover treatment abroad or you do not have health insurance you should buy a travel insurance which also covers health. Those are very common and not expensive and also people who have coverage through their home insurance or national system often have a health part to their travel insurance as it does cover more and gives peace of mind.
If you have a card on the European Health Insurance system, it does not matter whether it is a national system of health care or a system of private insurance, the insurance system will cover you on a comparable level.
In regard to England, you will be covered there by the card until Brexit, nobody can be sure yet what happens after. Buying a travel insurance is the safer plan.
If you do not have a health insurance at home, you might find that the health insurance part of your travel insurance has restrictions.
Living outside of Italy, I do not know whether you will be covered by the Italian health card, you can find contact details in the link, most likely in the external links at the lower part of the page.
People who need to apply for visa for the EU/Schengen countries always have to have a travel insurance which covers health to a minimum level.
If you do not have any health insurance I would advice you to buy an insurance with at least that level of coverage. You may have to pay for the medical treatment out of pocket but will get the money back when you send in the bills.
Regular health treatments are not covered by travel insurance but sudden pains coming up while traveling should be covered. In some countries all (travel) health insurances exclude existing conditions. So do check with the insurance company if you are covered in case of return of these pains.