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Yes - the US 501c3 entity may operate abroad - there is no issuesHowever the status of charitable organization will not be automatically recognized in the UK
To benefit from your organization must be recognized by HM Revenue and Customs (HMRC).
To get tax relief your charity must be: based in the UK, EU, Iceland, Liechtenstein or Norwayestablished for charitable purposes onlyregistered with the Charity Commission or another regulator, if this applies to yourun by ‘fit and proper persons’ recognized by HM Revenue and Customs (HMRC)You will find some details here
Similar to the US - donations to charity from individuals are tax free in the UK. You can get tax relief - see additional information herehttps://www.gov.uk/income-tax-reliefs/overview
In most situations you would simply create a whole owned subsidiary in the UK - and will apply for appropriate status for that subsidiary.That is how your US 501c3 organization would establish its presence in the UK.
Here is a small article regarding charities that allow tax benefits for both - US and UK - "dual qualified charity"http://www.lexology.com/library/detail.aspx?g=87590977-3d6c-462a-b72b-a6d763410edb