Now he is studying at UCL with a Tier 4 and I am... pregnant! Not the best timing to have our first child (hectic transatlantic move, can't do my job with this belly, Brexit etc...)but we are very happy!
You mentioned Brexit: does that mean you are hoping to use EU treaty rights of free movement, to be allowed to reside in the UK? If you are trying to use free movement to reside in the UK and are not working, how are you exercisng those treaty rights ?
As EU citizens, we can only visit another EEA country for 3 months using our own countries EHIC to pay for any emergency health treatment or use private insurance if it is not covered by the EHIC. A birth will not be covered by your Italian EHIC unless you suddenly go into labour when on a visit.
To continue to stay after those 3 months, we must be a qualifed person. We can start to be a qualified person at any time during that 3 months.
Free movement is about being a "qualified person" at all times, to have a "right to reside" in another EEA country. Cease to be a qualifed person or never be one and overstay that 3 months, then we lose the right to be in that EEA country. They can deport us back to our own EEA country and ban us from theirs for 1 year.
Just living in another EEA country is not being a qualifed person.
If you are not a qualified person who has a "right to reside in the UK", then you are not allowed to use the NHS for free.
With Breixt and the end of EU law in the UK looming, it is more important than ever that EU citizens make sure they are a qualified person and not just living in the UK when they have no right to. If there is any offer by the UK to let some stay, the safer bet is to make sure that you were allowed to reside in the UK in the first place.