As others noted, so far it can only be guesstimates. And for your PhD funding that does obviously depend on the small print, but:
There are other non-EU countries in Europe, such as Norway and Switzerland. My personal guess is that a brexited UK will have relations to the EU that are similar to those.
Wrt. to academia, a lot of the EU things are actually not restricted to EU countries. A number of associated countries e.g. for Horizon2020 are eligible as well (which are far more than just Norway and Switzerland). That are for example COST and Marie Skłodowska-Curie actions. Which would mean little if any change for a Marie Curie funded PhD if UK ends up being associated.
BTW, there seems (hearsay only) to be quite a distinction already in the UK between funded PhD positions that are eligible for UK citizens only and funded PhD positions that are open to EU citizens (or Horizon2020 countries) such as the Marie Curie funded positions.
As for loosing funding for a PhD position for which you do have a valid contract, I don't think the risk is high. The UK votes on leaving EU, not on abolishing their whole legal system. Retroactively changing requirements for temporary contracts would be rather costly in terms of violation of the principle of legality. (Though governments sometimes do have a tendency towards ex post facto rules like "as of last year, we'll collect a new tax")
As for the burocratic paperwork, sure there may be some changes. But again, my guess is that it wouldn't be much worse than the paperwork I had to do as a EU citizen when working in Italy as a postdoc (comparison: young worker exchange visum for Canada was less paperwork).