Council-run Kintore Way nursery in Grange Road - which was rated 'outstanding' by Ofsted earlier this year - faces having to make 15 staff redundant.
Members of the Southwark UNISON union presented a petition at Southwark's council assembly on Wednesday evening, having held a protest outside the council's Tooley Street offices in the hour before the meeting.
Susan Catten, schools organiser for Southwark UNISON, told councillors that the loss of 15 jobs "represents a catastrophic and devastating impact on early years SEND [special educational needs and disabilities] provision in Southwark".
Ms Catten said that 45 per cent of Kintore Way's pupils have special or complex needs, with the nursery becoming a "victim of its own success" as its reputation for SEND provision spread far and wide.
She continued: "The deficit the nursery has faced - forcing the governing body to cut posts - is not due to mismanagement, but simply because the funding protocol for nursery-aged children with SEND is broken."
Ms Catten called on the council to consider setting up a SEND resource centre at Kintore Way, saving some of the jobs and finding new ways to support pupils with additional needs.
She added: "It seems to me that Southwark is like a tale of two cities: London Bridge - with its affluence - and Bermondsey - with many of the poor and vulnerable in Kintore Way."
Responding to the petition, Cllr Jasmine Ali, cabinet member for children, young people, education and refugees, said she had raised the issue of nursery funding with Government, but the council was unable to write off Kintore Way's deficit.
She continued: "We want to help Kintore Way and it's absolutely imperative that we get the best support for our children with special education needs, and that's including children ... of nursery age."
"Nothing is off the table," she added.
"If I had the money I would absolutely plug the gap ... Please let's keep working together."
Later in the meeting Lib Dem councillor Irina von Wiese criticised the Labour administration for painting too rosy a picture of the council's provision for young people.
"Can you, Cllr Ali, really look the teachers, governors and parents of Kintore Way in the eye and say that everything is wonderful?" asked Cllr von Wiese.
"Is that how it would look to you if your son or daughter were told that their school or nursery may be closing or your job as a special needs teacher is being made redundant?
"I do think that we owe these parents, teachers and children a more honest account of what is happening in the borough."
South Bermondsey ward councillor Cassandra Brown (Labour) said: "I'd like to thank those from Kintore Way here tonight for all the work they've done for our community.
"I was almost moved to tears at a recent meeting with Kintore Way staff and parents, hearing testimonals of how much they've impacted the lives of so many children in our borough."
Cllr Brown placed the blame for the situation firmly at Westminster's door: "...it is the short-sighted Conservative government policy that is responsible for this crisis in our nursery schools.
"Under a Labour government - and a Labour council in Southwark - we should all strive to have real inclusion for our children."
The issues facing Kintore Way were highlighted at two Southwark Council meeting last month that heard moving testimony from parents and staff about the impact of the nursery's services.