Current EU law still fully applies and the UK hasn't made any major change to its immigration regime since the Brexit referendum.
As far as EU law is concerned, freedom of movement should apply until the actual Brexit happens, which should be two years after the UK triggers article 50 of the Treaty on the European Union. The British government stated plan is to notify the Council of its intention to withdraw from the EU on March 29, 2017.
Legally, two things could alter this timeframe. Firstly, article 50 provides that the treaties cease to apply once the withdrawal agreement enters into force. So the UK could leave the EU before March 2019 if an agreement is reached earlier. Secondly, the same paragraph also allows the EU to extend the deadline of negotiation if all EU countries agree. Both of these presently seem unlikely.
Furthermore, the UK could also conceivably cease to apply EU law unilaterally, for example by introducing some new restrictions or requirements on EU citizens' residence. I recall that some MPs toyed with the idea but this would force the EU to retaliate and make negotiations even more difficult so I don't think it's very likely.
After that, what will happen is even less clear. It's possible, although not particularly likely at this stage, that the EU and UK will enter some sort of Swiss-style association agreement with minimal changes to the immigration regime. It's also possible that it would create a completely different immigration regime but allow EU citizens currently in the UK to stay under EU rules, perhaps in exchange for reciprocity for the British citizens living in the UK.
In that case, something like the current rules would still apply to some people for many years. It's also conceivable that this would only apply during some sort of transition period or perhaps only for people who already lived in the UK before the referendum or some other specific date set during the negotiation. But all this is obviously very speculative, there is no way to know at this stage.