They claim that the few organists qualified to take up the job do not want to work for the Dean of Westminster, Dr Wesley Carr - the man who dismissed Dr Neary.
A firm of headhunters has been hired, but potential successors are refusing to throw their hats into the ring because of in-fighting and feuding at the Abbey, which has played host to many state events including royal weddings and funerals.
The paucity of candidates has prompted speculation that the post will be offered to Martin Baker, Dr Neary's deputy, who has been in charge of the music at the Abbey since last year.
Mr Baker, although comparatively inexperienced, has been highly praised, but is a Roman Catholic. Until recently, being an Anglican was a requirement for the job. This is no longer the case, but the appointment of a Catholic to the top musical post in the Church of England would inevitably stir up controversy.
"There's nothing more high-profile in the world than the Abbey," a senior member of the Royal College of Organists said. "But the extraordinarily uncomfortable atmosphere is putting a lot of people off."
Trevor Ford, editor of the Church Music Quarterly, said it is crucial that the right person be appointed, but added: "The appalling handling of the Neary case by the Abbey authorities has deterred a number of eminent musicians from applying for the vacant post. Engaging a firm of head-hunters to scour the world for Dr Neary's successor is seen by many as an utterly pointless exercise, as well as a waste of the Abbey's funds."
Dr Neary and his wife, Penny, an Abbey administrator, were sacked in April last year. Although cleared of dishonesty, a church court upheld the decision of the Dean to dismiss them. Respect for Dr Neary, combined with concerns about working with Dr Carr, who has been accused of being ruthless and over-controlling, are said to be at the root of the reluctance of potential candidates to apply.
The conductor Sir David Willcocks, former president of the Royal College of Organists and once an Abbey chorister, said: "If it had been a normal retirement at the age of 65 there would be no shortage of applicants. But I would be surprised if some people weren't cagey. My view is that the way in which things have been handled has made it much more difficult to get a successor."
Sir David was one of the renowned musicians who advised on appointments in the past but decided not to cooperate with the headhunters. It is understood at least one other former adviser followed suit.
A spokesman for the Abbey said they had waited until the Nearys had left their home in the Abbey grounds before starting the search in February.
Executive headhunters Saxton Bampfylde Hever had been brought in to ensure transparency and objectivity in the proceedings.