Why could interest rates go up if Brexit happens? I was reading this article, George Osborne says Brexit would drive up mortgage rates:


  George Osborne has issued a stark warning that mortgage rates will
  rise if Britain leaves the European Union.
  
  The chancellor said he thought it was likely interest rates, and
  therefore the cost of home loans, would rise if Britons vote to leave
  the EU in the referendum on 23 June. But Brexit campaigners accused
  Osborne of panicking and resorting to intimidating voters. 
  
  Asked if he thought the cost of mortgages would increase on a British
  exit from the 28-nation bloc, Osborne said: “The short answer is yes.
  I think that is likely, but I’m not in charge of interest rates.”


I was wondering what is the transmission mechanism for explaining why interest rates would have to go up if Brexit happens? Moreover, is there any historical example and precedent for this happening?
What would happen in the event of a "Leave" vote in the referendum? Well, the pound would quickly fall in value against its major trading partners - and some falls have already happened as the "Leave" vote appears to increase in probability. That makes imports more expensive, which is directly inflationary. Which pushes the Bank of England (the UK's Central Bank) to raise interest rates in response.

On top of that there are second-order effects which would pull in the other direction, but over a longer time period: there's the directly deflationary effect of the significant increase in unemployment. And the increased risk of further disintegration of the EU, with all the consequential risks to peace and prosperity that that would bring.

And as Alexis L. points out, the UK leaving the EU adds significant risk to the UK's currency; and that corresponds to a need for a higher return on the currency: that return is the interest rate. That's because if interest rates are too low, then people would keep selling the pound sterling, and it would keep dropping in value, pushing up inflation (requiring higher interest rates), and harming UK companies that rely on exports (creating political pressure to do something about the exchange rate - and that "something" is higher interest rates).