British citizens can continue to use their European Health Insurance Card - which entitles you to basic healthcare in all member states - until its expiry date.  Otherwise, you can now apply for the Global Health Insurance Card.  This enables you to get state in healthcare in Europe at a reduced cost or sometimes for free. 

Under this system most medical care in Spain is free at the point of delivery, although some items, like prescriptions, are not. More details here.

In France you have to pay up front, then reclaim what you have spent from either the French or the UK authorities.  More details here.

In Portugal most healthcare is state-funded. Please see here for more details.

Neither the GHIC nor the EHIC cover the cost of things like any private treatment, mountain rescue or emergency repatriation and we strongly advise that in addition to carrying the GHIC/EHIC you also take out a comprehensive travel policy including private medical expenses.

It is essential that non-EEA citizens have private medical cover, and again, we recommend that you consider a comprehensive travel policy to cover emergencies like mountain rescue.