The Galician Oilfields were indeed exploited by the Axis in WWII, along with the more famous Ploesti Oilfields, as well as the many around Lake Batalon. The Galician oilfields near Boryslaw were at peak production in 1909 though, with the Polish use of them being slightly inefficient in the post- Great War era. The oil production was over 2million tons in 1909 but only 822,000 tons in 1918 when Poland took over the territory from the crumbled AustroHungarians.
The Soviets proceeded to annex them during their occupation of Poland from 1939 to 1941, and then after Barbarossa, the Germans occupied Boryslaw. However frustratingly to the Germans, many of the specialists for drilling and refining escaped the June onslaught of the Wehrmacht, so production efficiency continued to decline.
Interesting side note from the main source I am using, Dr. Berthold Beitz was honored as Righteous Among Nations for using the production at Boryslaw as an excuse to save Jews during the holocaust.
The short answer: by World War II the Galician fields were essentially inconsequential because they'd both hit the bottom of their decline curves some 70 years after their discovery as well as being exploited and mismanaged during their glory years.
Galicia's fields have been subject to a bit of research in recent years given both their historic oddity - at their peak in 1909-1910 they allowed Austria Hungary to become the 3rd largest oil producer in the world after the United States and Russia - along with the cultural and sociological implications involving the Jewish owners and workers that either emigrated or were wiped out in the Holocaust. I'm going to largely skip the tale of the Galician Jews here because it's not my area of expertise, but it's a sad one and perhaps a flair in that area can provide a bit more color than I'll provide here.
But to give some context to this, Galicia - which nowadays is split between southern Poland and southwestern Ukraine - was a northeastern province of the Austro Hungarian empire that was massively poor, rural, and agricultural. Only about a quarter of the population was literate compared to 90% elsewhere, and infrastructure was substantially lacking. The western half was mostly Catholic Poles, the eastern half mostly Greek Orthodox Ukrainians, Yiddish speaking Jews were roughly equivalent to the two other populations and scattered throughout, and German speakers were a tiny minority. (Also, my national and racial terms wouldn't have been contemporary until after the First World War, but I'm using them for the sake of simplicity here.)
Galicia being one of the problem children of the Empire was an issue for decades - plenty of famines, not enough taxes, although unlike elsewhere at least not a lot of rebellions - but the most interesting aspect here about what resulted is that Austria paid very little attention to the province even after the discovery of oil and wax in the 1860s. This wasn't quite as negligent as it might sound; keep in mind that up until around the 1890s the major use of petroleum was for the brand new technology of kerosene lamps, in which between wax and refined kerosene Galicia became one of the major sources in the world. (Incidentally, this also led to devastation of the environment both here and in Pennsylvania, Baku, and Ploesti, since the other 80% of the barrel that wasn't refined into kerosene was just dumped into rivers and anywhere else readily available, with about the only upside being that production was a tiny fraction of what would become even a couple decades later.)
At that point, it was hoped that with the new industry perhaps the province could pull itself up by its bootstraps without much Imperial oversight or expense, so instead of the state controlling mineral rights (as was usual in Europe), they were generally allowed to remain with landowners. Why is this important? Because unlike in Ploesti and Baku - and ultimately Pennsylvania when John Rockefeller brutally consolidated control by refusing to refine oil that wasn't his in some form - the unintended effect of not having state control turned Galicia into a wildcatter's heaven that even Texas couldn't match a few decades later, with rights being sold in plots as small as 30x30 meters. Raising capital to drill an oil well in Galicia became one of the hottest investments in Belgium, in the UK, in France, and even in Canada; something like 40% of pre World War I ownership was foreign and another 40% from non-Galician Austria, and it was a major factor in allowing the Galician Jews to become their own mini magnates. (Amusingly enough, about the only time the Austrian government stirred was when Standard Oil started sniffing around for an entry point around 1910 and were promptly rebuffed.)
Unsurprisingly, this meant that oil production boomed since everybody wanted to get an immediate return on investment - and nobody at all looked at longer term issues like storage, transportation, or drilling wells that both aimed for smoother decline curves and that didn't ruin deposits with fairly careless drilling that allowed water seepage. This was much the same reason why the Pennsylvania oil fields dried up in a similar fashion, which was a couple decades earlier than they would have if tended more carefully despite Rockefeller eventually controlling refining and transport - by the time he'd completed his monopoly, the damage was already done. This was one reason Galician production peaked in 1909 at around 40,000 barrels per day and then declined to around 22,000 only 5 years later at the outbreak of World War I.
But there was another factor. Given Galicia's location and the known strategic value of oil by 1914, it was one of the first targets of Russia and was occupied in mid September 1914. The Austro Hungarian Empire eventually recaptured it in May 1915, but even then there were problems besides the destruction of about half of the infrastructure (outside of refineries, which were weirdly untouched - the British did a much better job ruining Ploesti before they pulled out) before the Russians retreated: western capital and technicians were no longer available, there was no efficient way to transport refined products to the Austrian and German Navies which desperately needed them (this is where the historic lack of state interest really bit them, since a pipeline would have been eminently possible prior to the war), and worst of all, many technicians that had fled the Russians were not allowed back into the region since Poles, Ukranians, and Jews were now all viewed as potentially subversive to the war effort as they sought various forms of self government away from the Empire.
The lack of technicians, consistent drilling, and equipment led to substantial damage to the fields and production dropped further to around 16,500 bpd in 1918. This was the first time the industry saw what's become a general rule of thumb that older fields will suffer a 25%+ permanent production decline if they're not consistently maintained - Iraq following the Gulf War is probably the most recent example - and it got worse once Galicia became one of the prime foci of the Polish-Ukrainian War after 1918 as both sides saw the fields as essential to their continued existence given oil was one of the few things both could sell for cash and foreign exchange. Not coincidentally, given the financing involved (they wanted to get paid back) France played a substantial role behind the scenes in tilting the new borders towards the Polish state, and most of the fields went to interwar Poland after they'd been damaged even further by going back and forth between the warring parties.
The other issue was that even without damage, these were still on the tail end of their decline curve anyway - the most relevant comparison was Pennsylvania's initial oil boom which also started in the 1870s and really fell off by the 1910s - and long past their glory days. Despite exhortation to investors about the 'limitless' oil fields and plenty of capital (foreign Western ownership increased to something like 2/3rds) and now technical support, throughout the 1920s the fields struggled to maintain their 1918 levels, and by the early 1930s had dropped to around 11,000 bpd, falling even further by 1940 to under 10,000 bpd once the Nazis had taken over. (To give you an idea of how little oil that is in modern terms, present day California currently uses slightly north of 1 million bpd with vastly more efficient consumption and there are a substantial number of wells nowadays that pump substantially more oil than that individually.) It was basically a drop in the bucket towards German needs, and despite an awful lot of effort to pull as much as possible out, that's why the Galician fields played a minimal role in World War II.
Sources: Oil Empire: Visions of Prosperity in Austrian Galicia (Frank, 2005), The Prize (Yergin, 1991)