Thebigsmith Hi, that's an unfortunate situation.
Is it possible to recover these manually by plugging it into a computer and making a backup of the whole phone or something
It depends...
Have you ever changed the default config in Settings ➔ Security ➔ USB Accessories? If the answer is no, then you will not be able to access your data.
Perhaps replacing the screen would solve it, but that is not a guarantee neither as the hardware and data may be corrupt from water damage.