Not a historian, but a physician who also enjoys the history of medicine and how things have changed over the years.
What happened then is the same thing that happens today if you do not, or can not go to the doctor. They either died - most likely from an infection overwhelming them with sepsis. Or with sincere luck (or prayers to the local deity) they got better, using whatever local pain relief they had.
In short, appendicitis is the inflammation of the appendix, typically from infection. The appendix was first anatomically described by Jacopo Berengario da Carpi gave the first description of this structure in 1522. Gabriele Fallopio, in 1561, appears to have been the first writer to compare the appendix to a worm. In1579 Caspar Bauhin proposed the ingenious theory that the appendix served in intrauterine life as a receptacle for the faexes. (5). When the appendix is perforated or ruptures, it releases the infection into the peritoneal cavity... which generally snowballs out of control without antibiotics and surgical intervention. John Parkinson was able to give a good description of fatal appendicitis in 1812. (5)
The first reported successful appendectomy was in September 1731 by English surgeon William Cookesley on Abraham Pike, a chimnney sweep.(1,2). The second was on December 6, 1735, at St. George's Hospital in London, when French surgeon Claudius Amyand described the presence of a perforated appendix within the inguinal hernial sac of an 11-year-old boy.(3). The organ had apparently been perforated by a pin the boy had swallowed. The patient, Hanvil Andersen, made a recovery and was discharged a month later.(4). ((NOTE: This can be a laparoscopic outpatient procedure today. It took the kid a month to recover. No anesthesia. No antibiotics.))
In older and more fragmented records, I’ve seen references to Egyptian papyrus, but no direct source and I do not read hieroglyphs very well.
In another article, “Aretæus of Cappadocia (30 AD) wrote, ‘I have myself made an incision into an abscess of the colon on the right side, near the liver,when a large quantity of pus escaped, which flowed also several daysthrough the kidneys and bladder, and the patient recovered.” is referenced at https://onlinelibrary.wiley.com/doi/pdf/10.1111/ans.12425 and seems to be an incision which drained the burst abscess and somehow healed. I don’t have time to chase that rabbit hole right now. Too much overtime.
A considerable share of the intestines cut off after a mortification in a hernia and cured". Medical Essays and Observations. Edinburgh: Society for the Improvement of Medical Knowledge. 5 (1): 427–31. 1742.
Selley, Peter (2016). "William Cookesley, William Hunter and the first patient to survive removal of the appendix in 1731 – a case history with 31 years' follow up". Journal of Medical Biography. 24: 180–3.
Yelon, Jay A.; Luchette, Fred A. (2013). Geriatric Trauma and Critical Care. Springer Science & Business Media. ISBN 978146148501
Amyand, Claudius (1735). "Of an inguinal rupture, with a pin in the appendix caeci, incrusted with stone; and some observations on wounds in the guts". Philosophical Transactions of the Royal Society of London. 39 (443): 329–336. doi:10.1098/rstl.1735.0071.