If the UK leaves the European Union on 31 October without a deal, EU citizens will, for a transitional period before the new points-based immigration system is implemented from January 2021, be able to move to the UK to live and work as they do now. If they wish to remain here after 31 December 2020, they will need by then to apply, free of charge, for 36 months’ European Temporary Leave to Remain or apply for leave under the new system. More information about these arrangements is available here:
UK nationals living in the EU at exit will be able to return to the UK with their family members for a period after exit under current arrangements based on EU law, including until 31 December 2020 with a spouse married after exit. More information about these arrangements is available here: https://www.gov.uk/government/publications/policy-paper-on-the-rights-of-uk-nationals-in-the-eu
Otherwise, EU citizens moving to the UK from January 2021 will, like non-EU citizens, need to meet the requirements of the new points-based immigration system.