A plea seeking the stay was filed by Father Dr Thomas Kottoor and Sister Sephy, the first and the third accused respectively.
Justice Sunil Thomas refused to stay the trial, set to begin in a special CBI court in Thiruvananthapuram.
The High Court also directed the CBI to produce the case diary, FIR and other documents related to the case.
Kottoor and Sephy moved the High Court after the special CBI court rejected the discharge petitions filed by them.
The CBI court, while dismissing the petitions on March 7, had observed that there was sufficient ground for presuming that the two had committed offences punishable under IPC Sections 302 (murder) and 201 (destroying evidences),read with Section 34 (Acts done by several persons in furtherance of common intention).
The court, however, had allowed the discharge petition of second accused Father Jose Poothrikkayil, holding that the prosecution had failed to bring out sufficient materials to proceed against him.
Abhaya's body was found in the well of the St Pius Convent in Kottayam on March 27, 1992.
Initially, the case was investigated by the local police and state crime branch, which concluded that Abhaya had committed suicide.
However, the case was taken over by CBI on March 29, 1993 following a legal battle by human rights activist Jomon Puthenpurackal.
The central probe agency in 2008 arrested Kottoor, Poothrikkayil and Sephy on the charge of murder.
According to the prosecution, Kottoor and Poothrikkayil were allegedly having an illicit relationship with Sephy, also an inmate of the convent.
On the night of March 27, 1992, Abhaya allegedly saw Kottoor and Sephy in a compromising position, following which the three accused hacked her with an axe and threw her into the well, it had said in its charge sheet.
The accused were arrested in 2008 and released on bail by the Kerala High Court a year later.