Rudd appointed Professor Alexis Jay to head up the inquiry, which is tasked together 13 separate investigations into one overarching review of historic child abuse in British institutions.
The inquiry was plunged into further crisis in recent weeks as its most senior barrister was suspended and others quit over fears the inquiry was too big and unmanageable.
Senior legal figures have called for it to be scrapped and refocused on identifying how to prevent abuse in the future.
Dame Lowell, who was appointed by Theresa May when she was Home Secretary, had been earning a salary of £360,000, plus £110,000 for an apartment in Kensington, another £12,000 to cover bills, and a Government car and driver.
Andrew Lavery, of the victims' group White Flowers Alba, told The Times the "galling" news came on the day the inquiry had invited him to attend a meeting in London but refused even to discuss meeting his travel expenses.
He added: "It's disgraceful but this inquiry has long been a shameful gravy trough."
Ian McFadyen, a campaigner, said: "A thousand pounds will give a survivor seven weeks of counselling and we're struggling to get those funds.
"I would have thought that someone who left without explaining why and without finishing the job would have declined any payoff."
A Home Office spokesman said: "The independent inquiry has a vital role to play in exposing the failure of public bodies and other major organisations to prevent child sexual abuse.
"We owe it to victims and survivors to get to the truth and the independent inquiry is continuing its vital work."
The inquiry has been beset by delays and controversies since it was first announced by the then home secretary Mrs May.
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.
May officially reconstituted the probe under Justice Goddard in March 2015 and placed it on a statutory footing, meaning it has the power to compel witnesses to give evidence.
The inquiry's terms of reference say that its purpose includes considering "the extent to which state and non-state institutions have failed in their duty of care to protect children from sexual abuse and exploitation". It covers England and Wales.
ONE ABUSE INQUIRY, 13 DIFFERENT INVESTIGATIONS
The Independent Inquiry into Child Sexual Abuse is to carry out 13 separate investigations:
• Allegations of child sexual abuse linked to Westminster - This will be an 'overarching inquiry' into allegations of abuse and exploitation involving 'people of public prominence associated with Westminster'. It will examine high profile claims involving 'current or former' MPs, senior civil servants and members of the intelligence and security agencies.
• The Roman Catholic Church - This will look into the extent of any institutional failures to protect children from abuse within the church in England and Wales.
• The Anglican Church - This investigation will look at the extent of any institutional failures to protect children from abuse within the Anglican Church.
• The internet - The inquiry will look into institutional responses to child sexual abuse and exploitation 'facilitated' by the internet. This will include investigations of the policies of internet firms.
• Residential schools - This will investigate abuse and exploitation of children in residential schools in both the state and independent sector.
• Nottinghamshire councils - This will look into the extent of any institutional failures to protect children in the care of Nottingham City and Nottinghamshire councils following allegations of widespread sexual abuse and exploitation.
• Lambeth Council - This will examine the extent of any institutional failures to protect children after allegations of abuse in children's homes run by the London authority.
• Lord Janner - This will look into allegations of child sex abuse against Lord Greville Janner. The peer, who died aged 87 in December, is alleged to have abused children over a period spanning more than 30 years.
• Protection of children outside the UK - This will scrutinise 'grave allegations' that have emerged regarding abuse by individuals working for British institutions and organisations abroad.
• Sexual abuse of children in custodial institutions - This will examine the scale of abuse within the secure estate for children and young people. The inquiry has identified Medomsley Youth Detention Centre, County Durham, as the first case study.
• Child sexual exploitation by organised networks - This will focus on institutional responses to systematic grooming and sexual abuse of children by groups of offenders as seen in cities including Rotherham, Rochdale and Oxford.
• Cambridge House, Knowl View and Rochdale - An inquiry into allegations of the sexual abuse and exploitation of children residing at or attending Cambridge House Boys' Hostel, Knowl View School, and other institutions where their placement was arranged or provided by Rochdale Borough Council.
Accountability and reparations for victims and survivors of abuse - This arm of the inquiry will focus on the support services and legal remedies available to victims and survivors of child sexual abuse.
