With the withdrawal agreement likely to take affect and the transition phase of the UK’s exit from the EU on the 31st January 2020, our Italian ministry friends have very kindly put together a few facts that we may need to know for our lives to continue as smoothly as possible in Italy, once the process commences.  Details are shown below:
ITALIAN MINISTRY INFORMATION FOR BRITISH CITIZENS REGISTERED IN ITALY AT THE TIME OF THE RATIFICATION OF THE WITHDRAWAL AGREEMENT.
Information on the status of UK citizens after Brexit in case of ratification of the withdrawal agreement is already available for all citizens and officials but we are certainly increasing our information campaign, in view of the last stages of the ratification of the withdrawal agreement and before the exit date, in order to avoid misunderstanding on your status.
As promised, here you find some information that can be published in case of the entry into force of the withdrawal agreement.
The source is the newly updated Brexit webpage on the Palazzo Chigi website ( http://www.governo.it/it/approfondimento/brexit/12733 ):
“UK citizens’ rights in Italy according to the EU-UK withdrawal agreement.
If the withdrawal agreement will be ratified before the exit date (31 January 2020), during the transition starting   from 1 February , in conformity with part IV of the withdrawal agreement and more specifically its article 127, unless otherwise provided by this agreement, European Union law shall be applicable to and in the United Kingdom during the transition period. In practice, it means that EU free movement law and social security coordination rules will continue to apply to UK nationals.  So despite the fact that UK nationals will no longer, as of the withdrawal date, be EU citizens, they will, during the transition period, enjoy the same rights as they currently do under EU free movement law. This means that nothing is going to change on 1 February 2020 with regard to free movement rights for UK nationals (and their family members) living in Italy. Likewise, UK nationals (and their family members) currently not living in Italy will be entitled, in exercising their rights under Directive 2004/38, to move to and take up residence in Italy during the transition period. 
After the transition period (starting from 1 January 2021 if not prolonged), the right of UK citizens and their family members will be governed by part II of the withdrawal agreement (with the exception of dual Italian/UK citizens who have never exercised their free movement rights). As stated by the preamble of the withdrawal agreement, the goal of the part II is to protect the Union and UK nationals as well as their respective family members, where they have exercised free movement rights before the end of the transition and to ensure that their rights are enforceable and based on the principle of non-discrimination while protecting also the right deriving from the periods of social security insurance.
Italy will provide before 31 January 2020 information on how to obtain the residence document after the exit date, in accordance with article 18 and with the conditions of Article 19 of the withdrawal agreement.  This document can be obtained by those UK nationals registered to the Italian Anagrafe before the end of the transition period (info on the Anagrafe in Italian language here:  http://www.governo.it/sites/new.governo.it/files/documenti/documenti/Notizie-allegati/governo/BREXIT/NODEAL_VademecumAanagrafe.pdf )
Official information on post-Brexit status of UK citizens in Italy is and will be available on line and in Italian language and you will be able to refer to it in a simple way, if needed.
The guidance note from the European Commission are available here (based on the November 2018 agreement):  https://ec.europa.eu/commission/sites/beta-political/files/2018-11-26_qa_citizens_rights_en_0.pdf
The European Commission will soon publish also an updated version”.
The British in Italy team will be following up with the Italian ministries during the month of January and will report back with any information.