I'm 13 I live in Anglesey and I was wondering if I can get my tongue pierced with my 18 year old cousin there and his permission?

No, your cousin has no authority to say OK. If they do it then they can be sued by your parents big time.

Most piercing places require parental consent for minors. A cousin is not a parent.
And why would a 13 yr old need their tongue pierced?

DO you even know the risks of having it done? There is a vein in your tongue that if punctured, is VERY hard to get to stop bleeding. My father in law is a medic and has seen this numerous times. Not only that, your tongue swells up for days, you can't eat, can't talk, etc... let me add one more thing, many people tend to click the piercing against their teeth and it ruins the enamel.