When B.J. is talking to the patient in the opening scene, a nurse holding a clipboard runs up and kneels next to the patient's head. When the camera angle changes, she is seen once again running up and kneeling down.
The British soldiers are wearing American dogs tags. British tags in the Korean War were round not tag-shaped.
While sitting with the patient addicted to morphine, BJ tells him "A few more hours and you can own your life again ". Even for the early 1950's it would be ludicrous to think that curing an opioid addiction could be a matter of hours.
Throughout the episode, Loretta Swit's fingernails can be seen at over an inch long. No nurse, especially a head nurse, would have fingernails that long.
When leaving the schoolhouse Klinger tells Father Mulcahey to "Throw it in to fourth gear!" Jeeps of the era had only 3 forward gears.
Ignoring the fact that aside from the commanding officer, all the English soldiers appear to have London accents (despite the regiment being the Gloucester Regiment) apart from Private Michaels who has a very obvious Australian accent despite his attempt to hide it.
When Klinger and Mulcahy are lifting the bell, it rings. But they were only lifting it (tipping it) to one side and the other was still resting on the ground. Under such circumstances, a bell will not ring (but will instead give a hard clunk if the ringer hits the side). For a bell to ring it has to be hanging free which will allow the bell itself to vibrate and resonate (when struck). But if the bell is grounded, even on one side, the ground will absorb and dampen the vibration of the bell (giving it, not a ring, but deadened clunk). Further compounding the effects of being grounded, Klinger (and at times Mulcahy) is also holding the bell and he too will absorb the vibrations of the bell (and therefore deaden the ring).
(at around 14 mins) When Father Mulcahy and Klinger are running back to their jeep, another vehicle is visible nearby. This is the exact same vehicle shown when they drive up to the shack. It hasn't moved at all.
At one point a British soldier declares that "Arsenal beat Manchester United two to nothing". No Brit would ever say "two to nothing", which is an Americanism. Always "two nil". It is surprising that Bernard Fox, himself a Brit, did not point this out.