The government has provided updated guidance concerning the status of EU citizens and their families in the UK after it leaves the EU on 29th March 2019.
The government claims that there is no need for EU citizens or members of their families to take any action at this stage. There will be an implementation period operating from 29th March 2019 until 31st December 2020.
After that date, EU citizens will be able to continue to be permitted to live here and have access to public funds and services. Later on this year, if already living in the UK, they will be able to begin applying for UK immigration status through new provisions referred to as the EU Settlement Scheme. Those who are living in the UK by 31st December 2020 will be able to apply for new status until 30th June 2021. After that date, they will have to either hold or have applied for UK immigration status to be able to remain legally in the UK. Citizens of Ireland will not be required to apply, although they will be able to do so if they wish to. It is anticipated that there will be like provisions for citizens of Iceland, Liechtenstein, Norway and Switzerland, although the details are still be negotiated.
The http://www.gov.uk website provides further details.
The precise details of the schemes for settled status and pre-settled status have yet to be approved by Parliament, but it is anticipated that obtaining UK settlement status under the scheme will enable you to continue to live and work in the UK. It will ensure continued eligibility for the use of public services (healthcare and schools), public funds and pensions.
It is anticipated that eligibility for settled status will be based on residence in the UK for 5 years. If you do not qualify for settled status because you have not lived in the UK for 5 years, you can be apply for pre-settled status, which can allow you to remain in the UK for a further 5 years. You will be able to live and work in the UK, with the same access to public funds and services.
You will then be able to apply for settled status once you have been resident in the UK for 5 years.
There is a Withdrawal Agreement that covers reasons for refusal to remain. As long as you make a valid application you will be granted either settled or pre-settled status, unless one of the following situations applies:
You weren’t resident in the UK by 31 December 2020;
You are refused on the grounds of having serious criminal convictions, for security reasons or for fraud.
If you have existing close family members (a spouse, civil partner, unmarried partner, dependent child or grandchild, and dependent parent or grandparent) living in the UK or overseas, they are also likely to be eligible for the scheme. Parents will need to apply on behalf of children. Any child born after their parent has been granted settled status will automatically become a British citizen if they’re born in the UK. It will not be necessary to apply for settled status on their behalf.
If you already have indefinite leave to remain, it will not be affected by the UK leaving the EU. Settled status gives some more beneficial rights and entitlements and you will be able to exchange this for settled status without having to pay any additional fee. If you are a non-EU citizen, it will be necessary to provide evidence of what your family relationship is to an EU citizen resident in the UK.
https://www.gov.uk/guidance/status-of-eu-nationals-in-the-uk-what-you-need-to-know