In a tactical-level wargame one would expect each nation's artillery to be governed by rather varied rules. The Germans get accurate artillery, but it's somewhat slow to come. The British get the fire very promptly. Their fire is less likely to seriously damage the intended target, but the effect of the barrage is going to be spread over a much wider area than a similar German or American fire mission. Furthermore for the most part, for the British and Germans, only specially trained Forward Observers can call in artillery fires.
The Americans of course get it all: Fast, deadly accurate (i.e., little or no drift), they get extra when they care (and even if they don't care), and they get the additional potent weapon of Time on Target. I should also mention that proximity fuses were introduced (sometime during the Bulge, I believe) so that Americans then can start getting the benefit of the far more deadly airburst fires (deadly to infantry and especially to open-topped vehicles).
Note that when in any kind of prepared defensive position, I would expect the British and German artillery to start responding as quickly and accurately as American. However, I suspect that the British and Germans may still have been limited to only FOs calling the artillery, even when in defensive positions.
Lastly, depending upon the period of the war, probably get a ton of not tremendously accurate, but decent artillery (again the amount and quality of fires will vary enormously during the course of the war). In a mobile situation they are probably largely limited to line of sight firing (maybe this is one of the reasons why they liked relatively big mortars so much?).
Another effect I might expect is differences in relative setup times for deploying artillery into new positions. British and American artillery should be fairly quick to get setup, being only really limited by having to set up the equipment, connect up their communications links, and lay in ammo. The Germans should require more time (unless all survey prep was done prior to the move) because the new position has to be surveyed, and the survey results have to be tied into previous results. Also the German artillery seems to have been rather more dependent upon wire communiations than were the Western Allies, and so may be further delayed because of this. I cannot speak at this point to what the likely limits are which affect Soviet setup.