Factual error: When the barracks are completely flooded, several of the trainees shove shower hoses down the toilets for an "unlimited supply of air" (as Merlin explains moments later, describing it as "simple physics"). However, that old fireman's trick is a technique for surviving dense smoke, which is not under tons of pressure. In the barracks flooded 9 feet deep with water, the pressure would be so enormous that it would easily gush straight down the toilets in a powerful torrent, eliminating any "unlimited air supply."
Factual error: When the barracks are completely flooded, Eggsy punches through a large two-way mirror to escape. The fact is that any glass (or plexiglas) thick enough to withstand many tons of water pressure without bursting would be as impenetrable as concrete to Eggsy's bare fist. He would need a chisel-tipped jackhammer to penetrate such a mirror.
Factual error: Every time the name of the Swedish princess is spoken, it is pronounced "till-dee." This is understandable for the majority of the characters, who speak English as their first language, but in the scene where the Swedish prime minister is interviewed about the disappearance, he also pronounces it "till-dee." He should have used the Swedish pronunciation, "till-deh."