To ostracise or isolate a part of the population due to inclement weather is indecent and indeed cruel in this day and age, hence a solution has to be quickly found to alleviate the hardship of anyone who wants to commute between the two islands.
Currently much is being written about the possibility of a tunnel but many problems of a geological and monetary kind have become apparent, with experts almost writing the project off.
I am no technical person and my suggestion might sound silly or only a basis for some fictional piece by a modern-day Wells or Verne, but I do think a simpler undersea route is feasible. Instead of digging a tunnel, lower a large pipe, made of sea concrete, precast on land and able to withstand high seawater pressure on the seabed.
During World War II undersea oil pipes were laid in the English Channel to relieve dependence on vulnerable oil tankers. This idea of mine involves a far greater magnitude but of much, much shorter distance!