Nonoo, 43, said she would undertake the role "first of all as a Bahraini" and that she was not chosen because of her religion.
Bahraini media had speculated over Nonoo's selection for the past few months.
Nonoo, a businesswoman and mother of two children, has served as a legislator in Bahrain's all-appointed 40-member Shura Council for three years.
Nonoo lives in both Bahrain and London and is the first Jewish woman to head a local rights organisation, the Bahrain Human Rights Watch.
She is Bahrain's third woman ambassador – the first being appointed to France in 2000 and the second to China.
Bahrain is a predominately Shia Muslim state but has Sunni rulers who are allied to the US.
About 40 Jews live in Bahrain, where there is one synagogue.
The authorities in Bahrain are currently planning to grant full citizenship rights to Jewish returnees, according to local reports.