I would bet money that there are already mobile phones on the island. My guess is that some guys from the island get out and work elsewhere and have acquired phones. Not sure they can get a signal on the island, but if they work elsewhere they'll have phones to make contact when they go to work.
I've been working with "indigenous" people groups for 30 years and they are always, always more connected to the modern world than our quaint idealism wants them to be. There are villages all over the world who have no electricity but have mobile phones. Someone sets up a solar charger or puts a car battery on a bike, charges up in town, and goes to villages with a charging service. And like I said, it's almost universal that young men will get out of the village, look for opportunities elsewhere, and bring their phones back home with them.
10-15 years ago in the Amazon, I started seeing signs appearing in remote tributaries with a drawing of a phone and an arrow. They were telling you that if you follow the arrow and go up the hill, you'll get a signal.
I could be wrong, maybe the Sentinelese are truly cut off from the modern world, but throughout the world the poorest of the poor and the most simple and remote people have mobile phones.