I would guess pretty much like modern soft/hard armor affect warfare today.
When firearms were introduced into the battlefield, armor (steel armor in this case) was still used. It continued to be used because early firearms were inaccurate, and inefficient. Most combat still took the form of hand-to-hand engagements with melee weapons (spears, halberds, etc.)
As firearms get better, production lines were established, mass production of accurate, affordable firearms improved, and armor was slowly phased out. Bullets penetrated steel armor. Charging armored knights couldn't survive against massed rifle/musket fire from a square of infantry.
Matter of fact, military armor was all but gone until WW1, when it was discovered that most military casualties in that war were caused by fragments from artillery shells hitting Soldiers' heads. Thus the steel helmets were introduced to stop this threat. Note that a properly aimed and fired rifle round, or even pistol round of that era (if close enough) could still penetrate those helmets. But the armor (helmet in this case) was designed to specifically counter artillery shell frags because those were the most effective casualty producing weapon at that time. Matter of fact, 56% of all casualties were caused by shell fragments, with machine guns close behind. Rifles and pistols caused relatively few casualties (a lot of factors go into this, but I won't get into it because it's a whole other discussion).
WW2 was almost the same. Armies are trained and equipped for the previous war, so they went into battle, mostly, with WW1 era personal equipment (in the beginning that is). We don't see widespread use of personal protective equipment other than helmets in any WW2 armies, aside from flak jackets issued to flight crews and sailors. The infantry pretty much only had helmets. As with the previous war, shell fragments (from artillery mostly), were the most effective casualty producing weapon of that war.
Matter of fact, we didn't see widespread use of extensive PPE until the Korean War, when the US started issuing flack jackets. Even then they were issued mostly as a defense against artillery fire, and not necessarily accurate rifle fire. (They could stop pistol rounds, but, pistols are almost useless in serious combat, thus pistol fire isn't really a worthwhile risk factor to consider).
It was only in 1996 that the US widely issued the Interim Small Arms Protective Insert (ISAPO) to be worn over the then issued PASGT armor designed to defeat small arms fire. But until the introduction of the Interceptor Body Armor, and the completion of that vest's issue throughout the Armed Forces, most infantryman did not have ballistic protection against rifle/machine gun fire.
Now that we're done with the real world history let's work on your question.
Due to the constraints you place on the Personal Protective Shield System (PPSS, pronounced the PISS by the troops), at first they will only be limited to certain units doing certain things that are of higher risk in encountering small arms fire than General Purpose Forces. In World War 2, these would be Raiders, Rangers, and other Special Operations Teams (SAS, SOE commando teams, OSS teams, etc). These units were expected to get into close combat with enemy troops in places where artillery fire was unlikely to happen (behind enemy lines, say). Some of these men will fall in battle, and ze Chermans will recover some functioning PISS units despite the precautions taken against this eventuality (troops were ordered to destroy their PISS before capture, for instance) and certain German units will also have PISS.
As the war continues, and as you say, mass production of PISS is perfected, more and more troops will have access to them. Their PISS wouldn't be as strong as the special folks' PISS, of course, but something is better than nothing, so you will see shields used in both sides of the war.
Tactics will eventually be adapted to counter this. The Germans would probably introduce semi automatic rifles much sooner, for instance, in order to facilitate rapid engagement against shielded troops. Marksmanship training would have emphasized on accurately hitting enemy troops multiple times with rifle fire in order to collapse their shields and kill them.
Assault tactics will also change a little as troops can now take more risks due to their shields taking some of the enemy fire.
Logistically, nothing much would change - except the curses thrown your way (or whoever was in charge) by logistics officers who now have several items added to their lists (PISS units, extra batteries, maintenance kits, rebuild kits, etc.). Armorers will now have an extra course to go through in their training (Shield Maintenance, Repairs, and Rebuild), or a Shield Technician MOS is added to the TO&E. With all the logistical nightmares that entails. (I kid. Mostly.)
Aside from that, nothing much would change. Massed artillery will continue to chew up troops (inducing slightly fewer casualties though, since some of those who would've been wounded by artillery fire, now escape unscathed). Tanks will continue to kill tanks. Firebombing would've killed about the same number of civilians. And the atomic bombs over Japan would've done the same, shield or no shield.
As technology advances, and we go into the modern era, I expect troops to still wear some form of armor under their shields (the Military is as risk averse as anyone who is closely under Congressional scrutiny), but like what we see today with armor, PISS will still mostly be a military issue. Insurgents will defeat the shields the same way insurgents now defeat PPE: Hitting troops where either the shields are weak, or multiple times to collapse them, or with a weapon of sufficient power to collapse the shield in one shot. IEDs will still chew up troops (I doubt that a shield will stop a hit from an EFP (explosively formed penetrator charge, or protect folks in Humvees from IEDs made from multiple 155mm artillery shells taped together). As before, the special folks will have better PISS than the general purpose folks (lighter, stronger, lasts longer!) and this technology will eventually dribble down to the line grunts.
About the only change is that newer generation logistical officers no longer curse you (or whoever is in charge) because now the PISS is in the TO&E so long, it's basically just another line item.