Letby, 35, originally from Hereford, is serving 15 whole-life orders for murdering seven infants and attempting to murder seven others.
Lady Justice Thirlwall said that lawyers for the former hospital executives - chief executive Tony Chambers, medical director Ian Harvey, director of nursing Alison Kelly and HR director Sue Hodkinson - had also written to the secretary of state for health to seek a suspension of the inquiry.
The findings of a panel of 14 international experts in neonatology and paediatrics were revealed by Letby's legal team last month.
Chairman of the panel Dr Shoo Lee said the experts had pored over trial transcripts and medical records and they "did not find any murders".
Those findings have been passed to the Criminal Cases Review Commission (CCRC), which investigates potential miscarriages of justice.
Letby's legal team hopes the commission will refer her case back to the Court of Appeal.
A letter written by Letby's solicitors to Lady Justice Thirlwall, seen by PA news agency, said a report based on the inquiry's findings would be of "little value" if the convictions were overturned.
It added: "It is likely that the CCRC will not take long to consider the application before referring it back to the Court of Appeal.
"It is now clear there is overwhelming and compelling evidence that Lucy Letby's convictions are unsafe.
"For the inquiry to be effective and the taxpayers' money not to be wasted, we urge that the inquiry be suspended and to wait for the outcome of the review to take place."
The CCRC previously indicated that a review of the application would take time due to the complexity of the case.