For Americans, any income tax paid to another country is offset by a tax credit on their US taxes, with carry over to future years, if it wipes out all of their current tax. It should be a wash for Phelps, et al.
But if he has good tax advisors, his intellectual property (trademarks, images) have probably already been transferred to a jurisdiction with a favorable tax treaty and Speedo wil probably pay him X for each appearance, then pay the licensing entity 5X for the right to use his trademark, images in connection with his appearance, thus removing much of the income from the jurisdiction of UK tax authorities.
Olympic stars to be clobbered by taxman
asia.eurosport.com/.../story.shtml