This is simply because the risk of a big fall in the pound should be weighing on the currency now, so if that risk disappears, sterling should rise.
We can roughly quantify this. We should think of sterling in the same way we should think of equities - as bets on different possible future states of the world. They are, in economic jargon, state-contingent securities. Their prices at any point in time should therefore be equal to the probability-weighted average of their prices in those different scenarios.
We know the probabilities. The odds against Brexit are around 9/4, implying there's a 30 per cent chance of it. NIESR's estimate of a 20 per cent drop in sterling implies that, in this event, sterling would be around €1.02. Given the current rate of €1.27, this implies that a vote to remain would push sterling up to €1.38. This is because €1.27 = (0.3 x €1.02) + (0.7 x €1.38). This is a rise of just over 8 per cent. We should expect a similar move against other currencies. This would take the pound back to its levels of last autumn. Obviously, if NIESR is too pessimistic about the fate of sterling in the event of Brexit, its rise in the event of a remain vote would be smaller.
You might think such a jump would be a deflationary shock. Certainly, it would reduce import prices. But there's a potentially offsetting effect. It's likely that some companies have delayed hiring and investment because of uncertainty about Brexit: MPC member Martin Weale said last week that such uncertainty "is the main reason why we expect Q2 growth to be weak". Insofar as this is the case, the ending of such uncertainty should trigger a pick-up in hiring and capital spending. This would offset the depressing effect upon output of the stronger exchange rate - especially because trade volumes aren't especially sensitive to moves in exchange rates.
Of course, if you're holding foreign currency-denominated assets, this rise in the pound would lose you money. It doesn't follow, however, that you should dump such assets.
Foreign currency is much like an insurance policy. It pays out well in bad times such as recessions or property crashes simply because sterling tends to fall then. Brexit fits this pattern; you can think of holding foreign currency as insurance against Brexit. But of course, the price of insurance is that you lose money in good times. Such losses might, however, be tolerable. A vote to stay in the EU might well see domestically-oriented share prices rise as uncertainty is resolved, so losses on foreign currency might well be offset by gains on some shares.
Whether this is a price worth paying depends on your individual circumstances: your exposure to foreign currency and to sentiment-sensitive stocks; your risk aversion; and your view on whether Brexit would be a good thing or not.
Now, you might object that all of this is based on a false premise. Maybe the Bank of England, NIESR and for that matter the majority of all economists are wrong and sterling won't fall much in the event of Brexit. Even if we vote to remain, however, this hypothesis will be tested. If sterling does not rise significantly in the event of a remain vote (or beforehand if the probability of Brexit falls) then mainstream economists will indeed have been mistaken. If nothing else, therefore, the referendum allows us to test the claims of mainstream economists against those of the Brexiteers.