Can anyone advise me. I have successfully installed Fallout 4 on my PS4 and when I'm due to start it tells me to press any button to continue but I am getting no response, hence I cannot start the game.

Press the button again, it doesn't register sometimes or you may need to uninstall and reinstall if turning your PS4 off and on doesn't work. Than again it might just be installing, it takes a while the first time.