will moving to a warmer climate permanently make my skin more tanned?!

You would tan but it would be slow, depends on your pigment. There are actually classifications throughout Europe of skin colour. Including *****, asian and with caucasian there are 3 or 4 types of white skin tone, being 1. fair like you, 2. Mediterranean Intermediate and 3. Dark, like North Africans/ Arabs.

I would be careful don't try tanning too quick, but do it safely and you'll be fine and even blonde haired people can go dark.

Once you move back to England, your skin would lose any colour had acquired. Skin renews itself constantly & without the sun to tan the new skin, you would go back to your usual colour.