The child protection expert will become the fourth person to lead the probe after the resignation of Dame Lowell Goddard last week.
Professor Jay previously authored a report which revealed that at least 1,400 children in Rotherham were subjected to sexual exploitation, by gangs of mostly Asian men.
Dame Lowell was appointed in April 2015 and had spent more than 70 days working abroad or on holiday during her time in charge.
An inquiry spokesman said the 67-year-old had spent 44 days in New Zealand and Australia on inquiry business and was entitled to 30 days’ annual leave.
Dame Lowell did not give full reasons for leaving but said accepting the job had been “an incredibly difficult step to take, as it meant relinquishing my career in New Zealand and leaving behind my beloved family”.
The inquiry has been beset by setbacks since it was set up in 2014 amid claims of an establishment cover-up following allegations that a paedophile ring operated in Westminster in the 1980s.
Ms Duckworth said: “It’s obviously a very frustrating situation, I’m sure not least for Justice Goddard.
“She did an incredible job, she has set an ethos which puts survivors right at the front of the inquiry.
“I’m frustrated by the situation, I don’t think anybody isn’t, but I think Justice Goddard gave up a lot, she came over to England from the other side of the world and she worked very hard, she was incredibly professional.”
Dame Lowell said: “The conduct of any public inquiry is not an easy task, let alone one of the magnitude of this.
“Compounding the many difficulties was its legacy of failure which has been very hard to shake off and, with hindsight, it would have been better to have started completely afresh.
“While it has been a struggle in many respects, I am confident there have been achievements and some very real gains for victims and survivors of institutional child sexual abuse in getting their voices heard.”
Home Secretary Amber Rudd said the inquiry would “continue without delay” and a new chairman would be found.
She described the inquiry, which has launched 13 investigations including strongly-denied claims linked to Lord Greville Janner, as the “most ambitious public inquiry ever established in England and Wales”.
Baroness Butler-Sloss stood down in July 2014 amid questions over the role played by her late brother, Lord Havers, who was attorney general in the 1980s.
Her replacement, Dame Fiona Woolf, resigned following a barrage of criticism over her “establishment links”, most notably in relation to former home secretary Leon Brittan, who died in 2015.