Appointed Chief of Staff of the newly recreated Luftwaffe in 1935, German Gernal Walther Wever was a major believer in the role of strategic bombing in the future of warfare, and it was his initiative that Germany undertake a project of research towards the creation of an arm of the Luftwaffe capable of long-range, heavy, strategic bombardment. A key aspect of this drive is the words strategic, unlike light and medium bombers which are usually confined to tactical bombing roles, oriented towards support of advancing infantry and armor units by localized attacks on enemy battlefield command and control, supply depots, and key road junctures heavy bombers oriented towards strategic bombing aim for major locations of production, transport, and communication deep in the enemy’s rear areas. In World War II the British and Americans utilized such aircraft in their raids both on Germany and (for the United States) raids upon Japan. Wever in 1935 wanted Germany to have the same possible strategic strike capacity however in his vision these weapons would be used in a war against the Soviet Union, to strike deep into the territory of the Soviet Union and destroy factories Stalin had relocated in eastern Russia.
Two such prototype bombers were ordered under Wever’s direction in response to a Luftwaffe competition to develop such a flying weapon, the Dornier Do 19 and the Junkers Ju 89, prototypes of both that flew respectively in 1936 and 1937. Both aircraft had teething problems, as all new developmental planes will, including power issues, flight characteristics, and design issues with hammering out the defensive weapons on both planes. However both prototypes showed considerable promise, with German experts slightly favoring the Junkers Ju 89 as a better aircraft, and both might with time have been able to successfully fly and matched the United States and British heavy bombers in capabilities. It is even possible that these aircraft might have been ready for use in limited capacity by 1940 had Wever been able to maintain pressure on the Nazi government, and the Luftwaffe, that these aircraft were a strategic necessity. Wever however died on 3 June 1936, before both prototypes were ready, in an airplane crash on his way to Berlin. His replacement, General Albert Kesselring (and possibly Goring himself, overall leader of the Luftwaffe), scrapped the heavy bomber project in favor of developing medium and short range bombers due to an anticipated focus by Hitler on short-range local wars. As well the German bomb sight technologies in 1936 and 1937 were not up to the demands of high-altitude accurate bombing, although by 1940 that problem had been solved.
The Germans did develop one long-range strategic heavy bomber, the Heinkel He 177, however it was plagued with design problems and challenges that Wever might have been able to stem off. The plane, as you will note in the picture above, had two propellers, to get the necessary power to successfully meet the performance requirements for the bomber it had to use four engines, two behind each propeller with the engines linked together. This was due to a shortage of powerful enough engines and a effort to design a very efficient aerodynamic design for the plane. However the plane was very heavy, due to its having to have major structural upgrades, so it could serve as a dive bomber. Dive bombers are usually small or, at most, medium aircraft with the goal that the pilot flies high over the target, plunges the plane over into a high-speed dive, and drops their bomb at a lower altitude to use the plane to guide it as precisely as possible on the target. The forces involved in the dive makes this sort of attack ideal for smaller aircraft, the Heinkel He 177 had to have major re-enforcing on its structures to take the strain of dive bombing. Plagued by these requirements, which Wever probably would have opposed had he been alive, along with the regular teething problems of a new aircraft, the Heinkel He 177 never really worked well and only few from 1942 to 1944, when its impact on the wars progress was minimal.
Now from a speculative perspective, in my opinion the reason this could have had a major impact on the course of the war was the Battle of Britain from 10 July to 31 October 1940, an air war, centered around a bombing campaign, conducted by Germany upon Great Britain to break its fighter support so the Luftwaffe could then destroy the British fleet if attempted to halt an invasion. Germany faced many challenges but one of the key issues was its bombers lacked the ability to carry sufficient bomb loads to do massive damage per plane, those planes were vulnerable to British fighters due to their need to operate at lower altitudes than a heavy bomber, and the fact that their range was shorter so they had less time over Britain. (German fighter aircraft also had range limitations which might have curtailed the usefulness of a heavy German strategic bomber.)
The intriguing aspect of the German heavy bomber was had it existed in 1940, in sufficient quantities to actually conduct a strategic war, it might have been able to directly bomb the British fleet repeatedly at its safe harbor in Scotland, Scapa Flow. The Germans did bomb Scapa Flow in 1940 using Junkers 88, a medium range bomber, but the planes could only carry a minimal bomb load to make the flight. Both prototype German heavy bombers, and the production Heinkel He 177, had roughly double the range of the Junkers 88 and the capacity to carry a far larger bomb load at those ranges. It is possible a proper strategic heavy German bomber force could have targeted, and badly damaged, the British fleet at Scapa Flow making the invasion of Britain by Germany in 1940 or 1941 more possible.
However the counter to that is the lack of range for German fighters, a problem the United States and Great Britain faced when bombing Germany from 1942 until 1944. Fighter coverage had to leave the heavy bombers behind requiring them to fly long distances relying on their own machine gun defenses, a tactic that caused heavy losses to the bomber fleets. This prompted the development of a long-range United States fighter to provide cover to the United States bombers for the entire duration of the flight. (The British switched to night time bombing and abandoned strategic precision to cut bomber losses.) German heavy bombers in 1940 might have been sliced apart by British fighters before significantly impacting the fleet. At the very least a reliable heavy bomber program for German in the 1930s might have given them another tool to use in their air war against Great Britain, another tool that might have shifted the course of the war.
But due to the death of General Wever in 1936, Germany abandoned the plan and history took the course we know.
Sources: Wikipedia articles on the Ural Bomber, Walther Wever, Dornier Do 19, Junkers Ju 89, Heinkel He 177, Battle of Britain, Scapa Flow, chapter in The Great Bombers on the Ural Bomber