In the UK TV cooking programme "Forever Summer" featuring Nigella
Lawson, they used some incidental music of a jazzy(ish) female
vocalist. Who is she? The (only) episode I watched was the next to
last in the series, presumably 5/6.

hi debbiem,
Sorry for the delay in answering your question. 
The next to last in the series was No. 7 (Green). I have contacted
Nigella's production company (Pabullum Media), and they inform me that
there were two songs played in the "Green" programme:
"Why shouldn't I" by Anita O'Day, and 
"Let there be love" by Julie London
I believe the person you are referring to is the jazz singer, Anita
O'day.
You can find out more about her from the Anita O'Day official website:
http://www.anitaoday.com/
hope this helps!