Mr May will officially take over the running of the trust in the autumn.
His appointment follows the departure of the hospital trust’s previous chief executive, Tracy Taylor, following ill health last year.
Rupert Egginton has been filling the role as an interim chief executive.
Mr May was first appointed as chief executive of the county council in April 2015, having first joined the authority in 2004.
The job is the most senior non-elected officer role in a council.
He had previously worked as the director for children, families and cultural services, and also as the deputy chief executive of the authority.
He said: “I am delighted to be appointed as the chief executive for the trust, which is very important to local people and the wider system.
“My focus will be on the current challenges facing the trust, including improvements in the maternity service, flow through the hospital, and the way the organisation is led.
“In the lead up to my appointment, I met a number of colleagues who work in the trust, and some of the patients who receive services.
“In all of these conversations, I was met by honesty and a commitment to improve.
“I intend to harness this commitment and work with trust board, staff, patients and partners, to ensure the trust is as good as it can be, now and into the future.”
NUH is responsible for both the Queen’s Medical Centre and City Hospital in Nottingham.
His appointment comes at a time when the organisation is under intense scrutiny over the quality of its maternity services, with the inspectors the Care Quality Commission (CQC) grading maternity it as ‘inadequate’ in 2020.
An ongoing NHS Clinical Commissioning Group review into the service, which is examining dozens of maternity incidents involving baby deaths and injuries, has been criticised by some families as an inadequate response.
Some families have instead been calling for an investigation led by midwife Donna Ockenden, who recently issued a scathing report into Shrewsbury and Telford Trust’s maternity care.
In a statement announcing his appointment, Mr May directly referred to the ongoing investigation and concerns of families.
He said: “I would like to reassure those who have come forward bravely as part of the independent maternity review, that I am determined that the trust provides full support to the process and can deliver the improvements needed for local mothers, babies and families.”
Trust leaders welcomed Mr May following confirmation of his appointment, while Coun Ben Bradley MP (Con),council leader, wished him well in his new role.
Nick Carver, NUH board chairman, said: “I am delighted that Anthony has been appointed as our new chief executive.
“He has extensive knowledge of the issues faced by the system, a proven commitment to partnership working and a strong track record in the provision of excellent services that are accountable to the public.”
Coun Bradley added: “Anthony has been a fantastic servant to Nottinghamshire and its residents over many years.
“Whilst I am very sad to see him leaving the council, I know that this exemplary service will continue in his new role at NUH.”