It would have been a matter of timing, first.
The US didn’t actually possess an atomic capability until Feb, 1945. This was when Truman was notified of the successful Trinity test.
So, the window of opportunity for the Germans to have used chemical weapons against civilian targets in the UK without an atomic response was before then.
After that Feb, 1945, Germany didn’t have the capability to deliver chemical weapons in any significant amounts against the UK to deter the use of atomic weapons against it. By the time the US had atomic weapons, the Germans had essentially zero strategic retaliatory capability. The Germans possessed no deterrent threat to prevent the US from employing the atomic bomb, had they chosen to do so.
The reason why the Allies didn’t use the bomb was because there was no reason to. By the time the bomb was a reality, Germany was defeated in all ways that mattered. German forces were surrendering in droves to the Western Allies, and most were continuing to fight the Soviets in order to gain an opportunity to avoid capture on the eastern front. Germany’s economy and strategic war-making capacity were both destroyed.
Before that time, well…
Consider that Hitler had chemical weapons at the start of the war. Their actual employment was technically possible from the moment the first shot was fired. Hitler chose to not only NOT employ chemical weapons against the British, but he also chose not to do so against the Soviets whom he considered true sub-humans.
Hitler was, himself, psychologically afraid of chemical weapons. He had experienced firsthand their effects during WWI when he, himself, was gassed. He also knew that had he employed chemical weapons, the British would have responded in kind. The British also had the technical capability to employ chemical weapons, and the military experience of their use in WWI was still fresh and relevant. Chemical weapons were well understood and created no irrational fear. They were seen as relatively ineffective on the battlefield against troops properly equipped. Their limitations as casualty producing weapons was not over-estimated.
Frankly, the window for Germany to use chemical weapons to gain any strategic advantage ended with the Battle of Britain. Up to the point it is just conceivable that indiscriminate chemical attacks against civilian targets might have caused enough domestic pressure for Britain to sue for peace with Germany.
However, I believe there were enough British WWI vets who knew and understood the basic ineffectiveness of chemical warfare beyond the initial surprise of first use against unprepared targets. After the initial surprise, far from calling for peace with Germany, I think the British would have been enraged and hell-bent on revenge. At that point, I have no doubt at all that Britain would have retaliated against German civilian targets with their own chemical attacks.
(Consider British public demand for retaliatory attacks against German cities after Germany first bombed British civilian targets during the Blitz.)
The net result of German first use of chemical weapons against British civilian targets might have been an even earlier entry into the war by the US which would have only caused the war to end even sooner. The psychological horror of any such German attack would have manifested itself even stronger in the US in favor of open intervention to assist the British.
