Central defender Davies was taken ill last week and had the operation on Saturday morning.
Neil says Davies’ absence will be a ‘minimum’ three weeks, possibly a bit longer.
“It’s basically the same kind of recovery time as a player having had a hernia,” Neil told the Post. “They have to open up the stomach wall and repair it back.
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“We are probably talking about three weeks minimum.
“Hernias are normally four to six weeks, so that can be a bit of a guide.
“It will be after the international break without a shadow of doubt.”
Davies was taken ill very suddenly last Wednesday night, having trained without incident earlier in the day.
Said Neil: “Ben was fine when we trained and went home as normal.
“Our physio Matt Jackson contacted me that night to say Ben was really unwell.
“In fact they had to call out an ambulance that night, which is highly unusual for a stomach ache.
“We took his bloods and his inflammatory response was really high.
“Then we sent him to see a specialist and by then it was starting to localise towards his appendix.
“He had to get it removed and had that done on Saturday morning.”
North End resume action after the international break with a televised home match against Bolton Wanderers on November 17.
That looks like it might come a touch early for Davies, as could the visit to Bristol City four days later.
The trip to Norwich on November 25 could be more realistic.
This has been Davies’ breakthrough season at Deepdale, so he will be frustrated to see it put on hold for the next few weeks.
He had started 11 games this season, 10 of them in the Championship.
Until Saturday, Davies had started seven games in a row, forming a steady partnership with Paul Huntington in the heart of the defence.
In the early stages of the season he had competed with Tommy Spurr to play next to Huntington.
Spurr himself has been in hospital recently, having an operation to repair a damaged medial ligament. His recovery is expected to take three months, with defenders thin the ground at present.