Getting an Italian passport

Hi, I'm in the process of trying to get an Italian passport, I was as a child on my Mothers passport (italian) and I have checked and found that I am still on their passport system. Has anyone done this and encountered problems? 

Also I want to try to get my children Italian passports, I am divorced (their mother is a British Citizen with no connections to italy), does anyone have experience of this and can advise on any problems I might encounter and how to solve them?

On a related topic, can I ask if anyone has encountered problems when using their EHIC card to visit a doctor in Italy? I had this problem this month when I was unwell in Italy and had to attend a doctor at the local hospital, they looked at my EHIC card and said it was not something they recognise (they also told me the UK is out of the EU and were reluctant to treat me, but they accepted my passport as ID).

Is there another form of ID that Italians must carry on them to identify themselves when seeking treatment at a hospital in Italy? If so can I apply to get one of those too?

Most people in Italy seem to think the UK is already out of the EU.

Your EHIC card only covers emergency treatment (broken leg, hospitalisation etc). So they may have been saying the treatment wasn't emergency so you weren't covered. They would want ID to enter your name and address on their computer system. If it wasn't covered by the EHIC then a bill may arrive.

Italians and foreign citizens who are permanent residents have a health card - "Tessera Sanitaria" that we have to show to get treatment. Even then it's not free as Italian healthcare isn't free - only emergencies.

It sounds like you are an Italian citizen however so it is worth getting your paperwork in order. If you don't live in Italy you can join AIRE which gives you certain rights. However, only residents get access to Italian healthcare as only residents pay into it. You need residency to get the Tessera Sanitaria. Therefore if you remain in the UK then the EHIC is the right docuemnt. What will happen after Brexit is an unknown at this point.

Looks as if some Italians already believe that BREXIT is in place, you will need to explain the situation to them. Your EHIC should cover you for emergencies or any other treatment which would not be ongoing, unless you have prior authority from the UK health department. Your EHIC alone is not enough, you ALWAYS have to show them another identity document, such as a passport. I went through all this when my husband suffered a stroke in Italy a few months ago. The Spanish EHIC covered everything, including ambulances and he is also an Italian citizen although we reside in Spain.
As for citizenship, as Admin says, you will have to contact your nearest Italian Consulate in the UK. You may need to get your Italian birth certificate, which should be in the system if you already found that there are records regarding your passport. Your children should be entitled to an Italian passport, but then, you will have to update your civil status records including your previous marriage and divorce certificates and the birth certificates for the children. Again, your nearest Italian Consulate should be able to inform you.