Yes, if you are a British citizen living in Turkey (or anywhere overseas) you still have the right to vote in the UK general elections and European elections. This right will cease to exist if you have been living abroad for 15 years and over.
Yes, as an American Citizen you can continue to vote, depending on the state in which you are eligible to vote.
Click the link for more informationhttp://turkey.usembassy.gov/voting_information.html
Disclaimer: The information on this site is provided as a general guide. Information published to YellAli is confirmed by official government departments in Turkey. Whilst every effort is made to ensure that the site is up to date and accurate, information may be subject to change at any time. YellAli does not accept any responsibility or liability for inaccuracies in this information.