Once they get to England, the French will do a Gallic shrug and it will be our problem. Once they're off the boat they will all claim asylum, and we'll have to process them and give them somewhere to live and something to live off.
However, when this was brought up before the referendum, it was, apparently, "Project Fear". Along with the Pound reaching parity with the Euro, the economy shrinking, foreign investment stagnating etc. etc...
About the only thing that wasn't forecast was a massive increase in racist and xenophobic incidents, but then the Brexiteers would have dismissed that just as blithely.