Can the Irish travel to the UK and use the NHS?

Not unless they are resident in the UK.

However, if they are visiting and fall ill unexpectedly, then they can receive treatment for free. (As can all EU citizens.)

If they are visiting I would expect so - that European Medical Health Card thing means you can get urgent medical treatment free anywhere in EU ( I think the public health service of the country of the patient reimburses the public health service of the country that provided the treatment).

If not necessary medically then no.  I expect this is the case rather than been certain,

Yes, wallflower but as I said, only if it is unexpected.

If you are resident in Ireland, you can't travel to the UK to get an existing condition treated unless you pay.

They can use it free of charge , nothing to do with the EU

A National health service is a fine idea, perhaps we can get back to having one soon.

Residents of RoI border counties go to Northern Ireland to visit GPs so they don't have to pay 60 Euro a visit.

I know that in Greece we ALWAYS have to pay a fee to see a doctor at any clinic, we never get this refunded and if it is possible to get it back it would involve effort. No Greek is asked for a penny when seeing a doctor here. I suspect this discrepancy is the same with every other country in the world.