If the song has been recorded and issued to the public, you DON'T NEED TO GET PERMISSION TO RECORD IT!

You do, however, have to pay the mechanical royalties for use of it, but that's fairly easy. Check with the publisher (listed on the record, or available through the record company) or a performing rights organization. I don't know who that would be in England, but some one here should be able to steer you to it.