Do I have to pay UK income tax? I know I get a tax free studentship but can I register for gross interest on a savings account?

yes you can claim gross tax on your savings. and you shouldnt have to pay tax on any teaching earnings either.

You should not take tax advice from an internet forum and should contact your local tax office directly for such information.

As far as we are aware it is the studentship, not the student that is tax exempt. However, if you are not earning any other money, it is unlikely that your bank interest would exceed your personal tax allowance and you should be able to reclaim any tax paid.