By the end of the 19th century, coal remained the king of transportation fuel sources. For ships and trains, the vast majority were still fueled by coal. The idea that one country's dominance of a resource's production would raise alarm in others, presupposes that that resource is a strategic commodity. In the period before petroleum became an adequate substitute for whale oil, and up until the late 19th century, both whale oil and petroleum were primarily used for lighting. While important for lighting before the spread of electric lights, oil commodities were not strategic commodities.

Strategic commodities are commodities that are important for a state's ability to defend itself and wage war. In other words, it is a strategic commodity if being cut off from it by your enemies would potentially cause you to lose a war. What really mattered strategically was having the material and fuel to supply your military, food for the troops and horses, coal for the ships and locomotives.

In the later decades of the 19th century, industrialized nations considered a number of commodities to be of strategic importance (coal, iron, saltpeter, etc.), but not yet oil. From the mid-19th century until the early 20th, many naval minds were becoming wise to the promise of oil as a ship fuel, but oil's emergence as a vital strategic commodity had to wait. By the time the British Royal Navy, through the influence of Winston Churchill as First Lord of the Admiralty, officially embarked upon the conversion of the fleet from coal to oil, submarines and some smaller ships were already using petroleum, and some navies had learned that they could temporarily boost their ships' top speed by spraying oil on the coals. However, the vast majority of naval ships remained coal fired until the Royal Navy started the switch in 1911.

Demand for whale oil had, by this time, declined sharply. Kerosene refined from petroleum displaced whale oil's market share very quickly through the late 1850s and early 1860s. Whale oil was simply too expensive and as soon as a cheap alternative was available, demand for it fell off a cliff.

So the period where whale oil was in demand and the period from which oil became a strategic commodity simply did not overlap, and were actually separated by about 50 years. This is probably why no country would have been too concerned about which other country was doing the most whaling.

edit: I would also add that not only was whale oil not a strategic commodity, but it was also a luxury good. Artificial lighting was a luxury in itself, but on top of that, whale oil was just the most luxurious of sources for lamplight among several cheaper options, including candles. So doing without whale oil wasn't just possible, it was quite normal for most people.