Could Brexit negatively affect PhD students in the UK who are EU nationals? This question relates to the potential withdrawal of the UK from the European Union, a.k.a. Brexit.

I am an Italian citizen, and I will start a (funded) PhD in the UK in September. 

Right now I don't need any particular documents to stay there, but I am worried that if the UK decides to leave the EU my position could be in trouble (especially with regards to funding).

What would happen in such a situation?

I am aware this question might be labeled as opinion-based, but I am looking for people who already got answers from their universities. I imagine financial plans have been established for the next two-three years already.
Update on 2016-06-28: The UK Government has put out an official statement on this issue.

The short answer is &quot;We don't know yet, but for a PhD program likely not.&quot;
The terms of Brexit have not been negotiated. They will may start to be negotiated if and after a vote to leave wins the referendum. (So that's the &quot;we don't know&quot;.) On the other hand, it will be at least two years after the vote before the actual British exit from the EU (presumably to negotiate and to allow a graceful transition). If you are starting a 3-year PhD program this fall, there's a good chance that the negotiations will not have concluded fast enough to directly affect your research.

Source: the BBC writes

If the UK left the EU would UK citizens need special permits to work in the EU?
Lots of people asked about this. A lot would depend on the kind of deal the UK agreed with the EU after exit. If it remained within the single market, it would almost certainly retain free movement rights allowing UK citizens to work in the EU and vice versa. If the government opted to impose work permit restrictions, as UKIP wants, then other countries could reciprocate, meaning Britons would have to apply for visas to work.
What about EU nationals who want to work in the UK?
As explained in the answer above, it would depend on whether the UK government decided to introduce a work permit system of the kind that currently applies to non-EU citizens, limiting entry to skilled workers in professions where there are shortages.

and

How long will it take for Britain to leave the EU?
This was a question asked by many people. The minimum period after a vote to leave would be two years. During that time Britain would continue to abide by EU treaties and laws, but not take part in any decision-making, as it negotiated a withdrawal agreement and the terms of its relationship with the now 27 nation bloc. In practice it may take longer than two years, depending on how the negotiations go.