The Government has, unsurprisingly, not given anything away in terms of what immigration rules will apply to nationals of EU member states beyond Brexit. However, things will inevitably change and it is difficult to see a scenario in which the rights of EU nationals to live, work and study in the UK remain the same.
It is unlikely, but not impossible, that the Government will seek to remove the rights of EU nationals who have already obtained the right to permanent residence in the UK. Less clear are the guarantees for those who have not exercised their Treaty rights in the UK for a continuous period of 5 years.
Given this uncertainty, the best option is always to apply to the Home Office to confirm whatever rights you think you might have, at the earliest opportunity. If you haven’t yet been in the UK for the continuous 5-year period required to obtain permanent residence, apply for a residence card instead. If you think you have passed the 5-year threshold to apply for permanent residence, apply as soon as possible in case changes are introduced in the way that the UK recognises the rights of EU nationals.
You can apply online (in most cases) or via a paper form. A European passport return service has recently been launched, meaning applicants can now apply online, then submit their documents at a participating local authority and have their identity documents checked without needing to send them off to the Home Office. This is particularly useful for those who travel regularly.
If you apply for permanent residence on the basis that you are or have been self-sufficient or a student, then you will need to provide evidence that you and your family members have held comprehensive health insurance during the period in question.
Assuming that they have been with you for the requisite 5-year period, you can include them on your application.
This is an interesting jurisprudential question arising from the 1992 Maastricht Treaty as well as interpretation of the 1969 Vienna Convention on the Law of Treaties and there has been a suggestion made in that British citizens could remain or register as citizens of Europe with a petition launched for this to be considered by the European Commission. The most likely scenario, however, is that there will be specific negotiation as to the rights of British citizens resident in Europe and EU citizens resident in the UK (which may hinge upon reciprocity) and as to the shape of any future free trade agreement (in which any continued rights of free movement are likely to be a central issue).
If you are married to a British national you could apply for a British passport immediately, provided that you meet the residence, language and Life in the UK requirements. If you are not married to a British national, you would need to have proved exercise of Treaty rights of residence for at least six years before becoming eligible for naturalisation.