According to the Nafziger books, an Artillery Battalion has an HQ Battery that includes a Platoon with a Signals Section, a Calibration Section and a Observation Section, as well as Staff Section and a Reserve Section. In addition the Battery has two Observation Platoons each with a Signals Detachment and an Observation Detachment. This if for 1937.
My understanding is, based on most other armies of the time, they all functioned in a similar manner.
The Staff Section would be told the fire mission for that day from the divisional commander. The Staff Section would then order the various batteries to their locations to support individual infantry regiments and battalions, working out with the Calibration Section where the guns needed to be in order to be in range for indirect fire. The Signals Section would then be tasked with setting up the relevent telephone communications. IF they were lucky, they would get some radio teams from the divisional signals units to improve communications. The Calibration Section would have teams that accompany the guns, to make sure they know on the map where they are, which is kind of important for indirect fire purposes.
Then, an Observation Section, or more likely, a team from an Observation Section, would be tasked to go to the infantry CHQ or Bttn HQ or Regiment HQ and use those signals elements to advise on fire missions that that CHQ/Bttn HQ/RHQ wanted during the course of the day's action, from the batteries allocated. This team would consist of an artillery officer/NCO, a signaller and perhaps a batman/driver. They would accompany the HQ they were supporting as they moved about the battlefield.
Thus, the role of the Observation Team was to facilitate the fire request that the infantry HQ wanted. This is important, as it is very different from a British/German/American FOO as most in the west seem to understand it, where FOOs were sufficiently trained and experienced to order fire, especially in the British army post August 1942.
I'll try and give a fictional example.
Let's say the 2nd Bttn 61st infantry regiment is due to advance next day, and overnight is assigned two batteries of 75/27 M06 in support. The Bttn commander allocates one battery to it's lead 1st company, and allocates the other to his control. An Observation Team accompanies each infantry HQ.
The 1st company advances and comes under MG fire, forcing them to a halt. The Observation Section have been reeling signals wire out behind with the CHQ signals section. Having identified the source of fire 900m away on a hill, the 1st company commander wants artillery fire on that hill. The Observation Team then tells him how to do it, allowing for things like wind, range, bearings, intervening terrain. Using the telephone or radio the commander then himself requests the fire, or gets the artillery team to do it. The request then goes through to the battery and they then fire or not fire, depending on whether they are in range, the telephone wire hasn't been cut, the batteries are working, the map reading is correct, the guns are actually available (have they been overrun, on the move, airstriked or counter-batteried already??). The Battalion commander then adds in his battery to the fire.
if they are not pressed for time, ranging shots would be requested first, this then gives the Observation team the chance to make the fire more accurate, then all guns in the battery are requested to fire, usually a number of shells.
In practice this is not the best use of the artillery, and it is more likely that the assigned fire support would consist of a pre-planned barrage in support of the advance. The attackers then have to guess where the enemy might be, or know from good overnight recce patrols, and plan a barrage accordingly. In this case example above, the 2nd Bttn 61st regiment has a fire plan given to it, the regimental commander having liaised with the artillery regiment commander the night before, and the batteries of 8x 75mm guns fire together a 20 minute barrage on terrain features in front of the 2nd Bttn, perhaps on that hill 900m. The fire plan might involve laying smoke, or barrages further inland on guessed enemy reinforcement points or known troop concentrations.
The third way of using the guns is of course, in the direct fire role, where the 75mm batteries actually move forward to the front lines and deploy to fire over open sights once the infantry they are supporting come under fire. Whilst more immediate response, and more accurate than indirect fire, it negates the range advantage of field artillery, and makes them very vulnerable to enemy fire. Even though the supporting ammo and staff trucks might be deployed 50m away from the gun line, in the open desert the battery could be knocked out with long range MG fire or mortar fire targeted on the trucks, leaving the guns immobile and virtually ammo-less.
This of course, is only my understanding of how it works, and if anyone can show difference please do.
This doesn't answer the original poster's question, about how observation teams are organised, nor how many they have, and whether the bigger guns were any different (105/28, 152/37, 149mm various, etc).
Mark