No.

The deficiencies of the Luftwaffe at Stalingrad, most famously the failure by the Luftwaffe to supply their encircled forces at Stalingrad, was not due to a lack of aircraft in Germany as a whole. It was due to a lack of dedicated transport aircraft and runways to fly them to and from, severe logistical problems, and because the Luftwaffe was primarily a tactical air force, not a strategic one.

First, even if Germany had 2000 additional aircraft it doesn't mean they could have used them at Stalingrad. They would require trained pilots, fuel, spare parts, maintenance crews, and air bases close enough for them to be in range of Stalingrad. All of these logistics were a growing problem for the 6th Army, and the Eastern Front in general, as a result of their pell-mell advance through the Russian Steppe towards the Volga and Caucus oil fields.

Hitler made matters worth by meddling with the plan. Instead of concentrating on the oil fields, he split off the 4th and 6th Armies to attack Stalingrad. Then when the 6th Army attack was going well he ordered the 4th Army back south. Armies do not turn on a dime, and the resulting traffic jam slowed the German advance by a week. When it was sorted out, Hitler ordered the 4th right back to Stalingrad! All this indecision and splitting of armies made the logistical problems even worse.

Unlike lighter and more nimble fighters, large, multi-engine, heavily loaded transport aircraft (and bombers pressed into the role) require long, finished runways for safe operations. Being heavily loaded their range would be reduced. Air dropped supply had not been practiced and often resulted in the supplies being scattered. The transports would have to land at a suitable airfield inside the pocket.

While the Germans held seven airfields inside the pocket at the start of the encirclement, just one was suitable for heavy transports: Pitomnik. It did not have sufficient capacity to supply the 6th Army. The Soviets knew it and worked it over every chance they could. Aircraft attacked the airfield and the transports. The approaches were covered with anti-aircraft guns. By January 15th it was in artillery range and two days later was captured.

The encircled troops required 750 tons of supplies minimum. The Luftwaffe was a tactical, not strategic, air force and had limited numbers of Ju 52 transports nor the organizational experience to pull off so large a sustained supply operation. Even with bombers pressed into service they could barely manage that. This lack of a true strategic capability would bite them both in the Battle of Britain, and in their lack of ability to strike at the Soviet rear.

In addition, many of the types used in the Battle of Britain were no longer front line combat aircraft, or did not have the range to operate over the very long distances in the Eastern Front. The Bf 109 E was being replaced with later variants, plus the superior Fw 190. The long range destroyer concept of the Bf 110 didn't pan out and it was increasingly relegated to reconnaissance. The Do 17 was being replaced with the Ju 88.

The most important aircraft from the Battle of Britain at Stalingrad would be the He 111, pressed into service as a capable fast transport. The other bomber pressed as a transport, the He 117 was introduced in 1942 two years after the Battle of Britain.

The loss of 3000 experienced air crew was of larger concern than the loss of aircraft. Germany, planning for a short war, did not have a good system in place to train new pilots. Experienced pilots were not rotated back to train other pilots, they had to fly til they died.

No. With more aircraft Germany would have had the chance to reduce Stalingrad to debris. But debris are actually a good place to defend, even more than a city no being bombarded.
In the city itself, Russian tactic in Stalingrad consisted in reduce as much as possible the distance with germans, to prevent german bombing since both sides were too close.
More aircraft would have been useful only to bombard artillery positions in the easter bank of the Volga. But not to eliminate infantry in the city. Hence, the only available effect of having more aircraft would have been less deaths of germans during the first stage of the battle (before the counterattack), due to less russian artillery fire.

Germans mostly needed transport aircraft in the last stage of the battle (when they were already encircled), but these kind of aircraft was not used against United Kindom. I also assume that reconnaissance would not change (to discover on time soviet preparations), because that kind of planes were not used in Battle of Britain.

Finally, as a reality test. Without the Battle of Britain, germans would have invaded the Soviet Union earlier. But this is fiction. So everything is different in that alternative reality.