Everything else would be the same in the war: (no Me-262 until 1945, no atom bomb, and the Pearl Harbor attack still happens.)

I know the Tiger II's engine was inefficient and underpowered, but we can assume that later in the war Germany would have fixed this problem with diesel or gasoline fuel.

No, the issue wasn't with the tanks themselves, although they definitely had their fair share, the issue was quantity versus quality.

The Tiger II; a more beautiful tank has never existed

In 1939, the Panzer Corps were already a significant threat on par with French Tanks which were thought to be the best in the world at the time, the implementation of the Panther and Tiger II would only serve as overkill. The areas where the Panzers failed was the Eastern Front where they were pitted against the inferior yet numerically superior T-34. The T-34 was a good all-around tank but lacked the necessary punching power that the Panther and Tiger could bring to the table. The Soviets overcame this with an age old Russian tactic; numbers.

Germany was never close to matching the Soviet Union in terms of production

Simple; in the 1920's and 30's when the USSR was undergoing massive industrialization, they decided that all civilian factories should be able to convert to military production in times of war in a month or less. Germany had not done this, it had designated factories for civilian production and military production. It was only in 1942 when Albert Speer took over as Minister of Armaments and War Production that he performed what is seen as a "miracle" today, increasing Germany's military production every year by incorporating civilian factories and slave labor until their defeat in 1945 despite constant Allied bombing campaigns and draining economic resources.

The simple fact is that the Panther and Tiger would have made little difference in 1939 as the Germans were already curb stomping through Europe with their Panzer III's and IV's. The inclusion of the advanced medium Panther and the heavy Tiger II would only serve to cripple the German economy sooner than if they had been implemented in 1944. Everyone tends to forget that these monsters were money suckers, expensive to build and difficult to maintain. In contrast, the T-34 was a rugged and reliable war machine that required little maintenance (except early versions which tended to break down after about 200 km) and was very easy to build (the Dzerzhinsky Tractor Factory was famous for having T-34s roll off the line and directly into battle during the Battle of Stalingrad).

Tank costs for each nation's main tank; note the Tiger II's astronomical price compared to the T-34

If Germany wanted to win the war in the East, they needed numbers. Quality can only hold for so long before the sheer quantity of your opponent overwhelms you. Germany didn't have the industrial capacity that the USSR did and it was only a matter of time before the German's were going to be pushed back on the Eastern Front. The only chance Germany had was either to cut off the industrial capacity of the USSR before it shifted to the Urals or design a cheaper and more reliable tank.