Before moving to Hungary from the UK you need to make sure that you apply for a European Health Insurance Card which has now replaced the former E111 form and several other healthcare forms.
Application forms are available either at a UK Post Office or at the UK Department of Health website www.dh.gov.uk/en/index.htm. This is a very simple form to complete and once you have received your healthcare card it entitles the holder to healthcare at any hospital throughout Hungary at a reduced cost or free of charge in some cases.
An EHIC will only cover you for emergency healthcare treatment in Hungary. Remember though that this really is just for tourists, but will ensure that you do not end up having to pay for treatment during your first few weeks or months in Hungary.
Private health insurance is widely available in Hungary.