Now Brexit has taken place, current European Health Insurance Cards (EHICs) which give the right to free healthcare in the EU continue to be valid. When they expire a new Global Health Insurance Card (GHIC) can be applied for on the NHS website. https://www.nhs.uk/using-the-nhs/healthcare-abroad/apply-for-a-free-uk-global-health-insurance-card-ghic/
Under a new agreement with the EU, the GHIC offers equivalent protection to access medical care when the current EHIC expires. The NHS website states applications take 10 days to process. Each family member travelling requires their own GHIC, including children under 16. For more information on the GHIC, see the NHS website.
Although the GHIC confers a right to medical treatment in the EU, it is not a substitute for travel insurance.