There's a lot of detail that I can't really go into (I'm on a mobile), but here's the general breakdown on medic treatment:

• The Allies tended to respect the Geneva Convention pretty religiously, and their forces tended not to shoot up ambulances, hospital trains, medics, anything with a big Red Cross on it.

• I can't remember reading of seeing anything about Japanese battlefield medics in the Pacific, but I'd assume (at least initially), Americans would have respected their roles.

• The Japanese were by and large monstrously cruel to their opponents; they'd wound GI's, let the screams draw a medic or stretcher party, and shoot them all. Wounded Japanese would let medics approach them, and then roll over to reveal a live grenade. So on and so forth. I can assure you they wouldn't respect anything the Chinese had in the way of medical personnel.

• The Germans (and by extension Axis) had a pretty robust medical system, with lots of training, field personnel, casualty stations, etc. They tended to respect the rules surrounding medics as far as the Americans/British/French/Canadians were concerned.

• All bets were off on the Ost Front; Antony Beevor's Stalingrad goes into detail about how both sides would shoot medics, kill all the wounded in a captured hospital, etc. there were official policies on both sides about treating the wounded fairly, but that was mostly lip service. The Nazis viewed Slavs and Russians as subhuman, and acted accordingly. Beevor's book also mentions medics tossing away their armbands, and German hospital transport planes regularly being shot down by Red Army pilots.

I cannot stress how general my reply is. Every bulletpoint above has tons of corroborating evidence, but also lots of anecdotes where people acted contrary to what I wrote. This is a broad, freshman-level college type of answer that's factually correct, but not nuanced.

I don't know about German medics but I do know a little bit about British medical personnel.

During World War One, there weren't really a great deal of medics on the front line. Usually, they would be behind the line in the reserve trenches or even further back in a dressing station. After all, these people were invaluable for the hundreds of soldiers wounded daily. A lot of the time, it was the job of ordinary soldiers to patch up the wounded and stretcher them off to an ambulance or field hospital. During a large-scale attack, there might have been medics brought up in preparation, but for the most part, there weren't many large-scale battles anyway. When a soldier was wounded during downtime, it was usually by sniper fire or a stray artillery shell. If the soldier wasn't dead, he probably would have been by the time he reached any medical personnel. It was important that normal soldiers could at least perform basic first aid (much of which was learnt on the battlefield), if not create a tourniquet and bandage up more terrible wounds.

With this, there weren't many instances when a medic would run into a German soldier. Recovery crews that were sent to retrieve injured men or bodies from the battlefield were usually ordinary soldiers. Even stretcher-bearers were usually not medical personnel. I've not read a story where a German soldier has avoided shooting a medic, but there are many, many stories of German soldiers not shooting their enemy, so it is certainly likely that it did happen. Considering the many stories of humanity during World War One, it wouldn't surprise me.