Bishop, who died in January 2022 aged 55, was jailed in 2018 for a minimum of 36 years after being convicted of the murders of Nicola Fellows and Karen Hadaway, who were both nine when they died in 1986.
He had been cleared of their murders at an earlier trial in 1987.
Jennie Johnson was given a six-year prison sentence in May 2021 after a judge and jury heard that she had lied at Bishop’s original 1987 trial.
On Wednesday, three judges dismissed the 56-year-old’s appeal bid by after considering arguments at a recent Court of Appeal hearing in London.
Bishop was 20 when he sexually assaulted and strangled Karen and Nicola in a woodland den in Brighton, East Sussex, in October 1986, judges heard.
He was cleared of their murders in December 1987, but within three years went on to kidnap, molest and throttle a seven-year-old girl.
While serving life for attempted murder, Bishop was ordered to face a fresh trial under the double jeopardy law, following a DNA breakthrough.
Jurors convicted Bishop, following a 2018 trial at the Old Bailey, on the 31st anniversary of his acquittal.
The appeal judges heard that Johnson was jailed by Mr Justice Fraser at Lewes Crown Court in May 2021 after being convicted of perverting the course of justice and making a false statement.
Johnson said she had been “a vulnerable young mother” in an “abusive relationship with Bishop” in the late 1980s.
Lawyers representing her argued that a fair trial had been impossible more than 30 years after the “relevant events” and said proceedings should have been “stayed”.
They said the judge’s summing up had been “demonstrably one-sided” – and Johnson had been “denied a fair trial”.
Lawyers also said Johnson’s “defence of duress” failed “adequately to address” the circumstances of the “violent, coercive and controlling relationship” she had been in with Bishop.
The appeal judges disagreed and refused to give Johnson “leave to appeal”.