Open the iTunes application that is saved on your computer. Go to the iTunes website and download the program if you do not have it already installed. This program is required to be able to buy downloads from the Apple Store.
Type in the name of the British artist or song you would like to download at the "Search Store" bar. This is located on the upper right hand corner of the screen.
Click on the "Buy Song" link to buy singles and the "Buy Album" link to buy the entire album. Singles range from .99 cents to over $1 and albums range from $7.99 to over $16.99.
Enter in your email address and password then click "Buy." Confirm that you would like to buy the song by clicking the "Buy" link once more.
Wait for the song to finish downloading and then listen to it by clicking on the folder that says "Purchased" on the left hand tool bar. Your payment method will then be charged for the order.