In answer to Dunedin, here is an example of a charity that makes tax relief possible in both countries simultaneously. They have organized things quite carefully, as explained below.
http://www.cantab.org/about-us/cam-uk"If you are subject to tax in both the UK and US, you now have the opportunity to give to Cambridge in America (UK) Ltd, a registered charity in England, and be simultaneously eligible for tax benefits in both countries.
Although registered in the UK, the charity’s sole shareholder is Cambridge in America, a US charity. This structure enables those who are liable for both UK and US tax to obtain tax relief in both countries on the same contribution."
CAF America has a similar structure.
https://www.cafonline.org/my-personal-giving/international-giving/dual-tax-payer-donations.aspx"If you are an individual paying tax in the UK and US, you could find that the US tax rules invalidate UK tax breaks on charitable giving and vice-versa.
Through our CAF American Donor Fund* you can:
Claim Gift Aid in the UK, increasing the value of your donation by 25%
Claim UK tax relief (at 40% or 45% rates of tax) and get a tax deduction in the US.
*CAF American Donor Fund is the business name of Southampton Row Trust Ltd, a UK-based charitable company registered with the Charity Commission in England and Wales. The fund is wholly owned by CAFAmerica, a US 501 (c)(3) public charity."