There would have to be a regency in the event that the new king or queen is a minor. Although the British Monarchy has declined in power over the last few centuries, it does still fulfill several important, if not symbolic, functions.
So lets say that the Queen Elizabeth, Prince Charles, and Prince William are all taken out. The next in line for the throne would be Prince George, who is currently two years old. As I can't see a toddler giving the annual Speech From The Throne for the next several years then a regent would have to be appointed. Parliament would most likely pass a bill appointing an individual to act in King George VII's (if that is the name taken) name until he reaches adulthood.
I would've looked for Prince Andrew, the Duke of York, to be the regent due to his age and experience, but he's had some scandals occur which may take him out of the running.
Prince Henry may be a good alternative, but his youth may play against him. Regardless, I'm sure that he would have some say in the Regency and he would most likely become and important figure for the new king.
Perhaps a senior member of the House of Lords would be selected if there were no viable alternatives. That or perhaps a collective regent made up of government officials.
Oddly enough, this may actually give the nascent Republican movement an argument for their cause. I could see their leaders saying something like: "If our head of state was elected, then we would have a new one immediately rather than waiting 16 years for one." Of course they would have to wait a while to make their case as the public would still be in tremendous shock and grief for the loss of the three senior members of the Royal Family.