UPDATE 19 December 2019 Living in the EU preparing for Brexit:
UK nationals in the EU
The Withdrawal Agreement
The Withdrawal Agreement sets out the terms of the UK’s withdrawal from the EU and provides for a deal on citizens’ rights.
You will be covered by the Withdrawal Agreement if you are a UK national lawfully residing in another EU country at the end of the implementation period, on 31 December 2020.
Residency documents and status
You and your family may need to apply for a residence status to confirm that you were already resident in the EU country you live in before 31 December 2020. You will have until at least 30 June 2021 to do this.
The EU country where you live will set up a system for applying for a residence status. The application will be short, simple and either free of charge, or cost no more than applying for a similar document, for example a national identity card or passport.
You will have until at least 30 June 2021 to submit your application information on how to apply will apply in UK government’s Living in guides when it is available.
UPDATE 11 March 2019 Spanish Goverment Publishes No Deal Contingency legislation and announces if UK leaves without a deal there will be 21 months to comply with new procedures required of British Nationals living in Spain (dependent on UK reciprocating)
…. In the event of no deal UK nationals living in Spain will maintain their legal residence status after 31 May 2019 or earlier in the event of a deal and they, and their family members, will have at least until 31 December 2020 to get a (third country) Foreigner’s Identity Card (TIE). Further information on the process for obtaining this card will be released in due course check here as and when availablesource:https://www.boe.es/boe/dias/2019/03/02/pdfs/BOE-A-2019-2976.pdf
Latest information on Health Care issues (S1s and EHIC cards) as they relate to Spain can be found here Page entitled SOCIAL SECURITY, HEALTHCARE AND HEALTH PROFESSIONALS http://www.lamoncloa.gob.es/lang/en/brexit/howtoprepare
I can’t emphasise enough how important it is for British Expats who live here in Tenerife to make sure they are not only registered, but CORRECTLY registered on the Central Register of foreigners and have a Green Certificate of Residence Registration (Also wrongly called Residencia by some people) This includes obtaining DUPLICATES if there have been changes of address, or circumstances, or if the original has been lost or stolen. This also includes children who are not registered even if their parents are. You must also make sure that you are registered for Spanish State Health Care if you have the right to do so. This might include so far unregistered dependants of workers or self employed for example. Getting this done is cruicial and due to the agreed extension you may have at least until the 1 Nov 2019, or possibly sooner depending on the outcome of the Withdrawal Agreement and Political Declaration on the future relationship between the UK and EU. You can stay up-to-date with official information on the most recent developments from the British government here and from the Spanish government here