Rio bronze medallist Tai, 18, lowered the S9 100m backstroke record after touching in one minute 7.66 seconds to win gold in Sheffield.
She also took bronze in the 100m butterfly, with triple Olympic gold medallist Firth, 21, breaching the S14 mark in the same race (1:4.62).
Both swimmers were competing in mixed category races.
Tai, who took seven-tenths of a second off her best, said: "It seems so surreal.
"I've not been close to my PB in a while so to knock lots of time off it and get the world record is crazy."
Home hopes Alice Tai and Bethany Firth both set world records at the British Para-Swimming International Meet.