The four-day-old girls, who had been joined at the liver, are in intensive care at Great Ormond Street children's hospital, where a spokeswoman said they had spent "a peaceful day" following "a very settled night".
The girls, who come from the Bristol area, lie side by side on ventilators while they recover from their three-hour operation which took place on Saturday.
Their parents now face an anxious wait to learn whether the girls' encouraging first 36 hours will develop into a full recovery. The hospital refused to comment further on the girls' chances of survival, saying only that it was "early days after a very major operation".
"They are both still poorly but stable. They are still at a very critical stage and are being monitored carefully," the spokeswoman said. "We are looking forward to a second settled night."
The twins were "unstable" during the operation at the London hospital and needed "external cardiac massage" to restart their hearts. Now they are receiving round-the-clock care from a team of specialist doctors and nurses.
Their father said: "We have been worried through the pregnancy, but during the operation we felt very confident in the surgeons and that everything possible was being done. We are now very anxious - waiting to see how the twins get on is very, very difficult."
A team of surgeons and anaesthetists carried out the emergency operation after the twins - who were born on Thursday - were airlifted to London from St Michael's Hospital in Bristol by a RAF helicopter on Friday afternoon after it was found that they were suffering from peritonitis.Reuse content