It does say just "WMD", I think a more interesting discussion is the possible Nazi development of the atomic bomb, poisonous gases and such wouldn't have changed the war drastically enough.
IDK.
Part of the problem is that chemical weapons would increase the effect of bombing civilians and logistical stresses. I see the deployment of chemical weapons playing lightly in favor of the allies in total:
Germany's initial use of Chemical Weapons might really help them on the Eastern Front--but even if Germany managed to win at Stalingrad as a result and spare a third of a million men, its logistical stresses have intensified and the Red Army can be counted upon to respond. Indeed, the Red Army might face problems of having little gear to protect against chemical strikes. But the Wehrmacht and its conscripted Ostruppen are not going to be much better prepared.
In the West, the USA is going to avoid any chemical strikes entirely, and while the UK might suffer some nasty attacks, its going to be Germany itself that faces the worst abuses. The USA can make the weapons but never face the dangers of the attacks. The UK might face a problem with V2s and nerve gas, but unless this is happening in 1940 its really Germany getting screwed.
Can Germany really spend the extra resources building chemical weapons and protecting themselves from their effects? Sure, but somethings got to give. Now, it would be nice if Germany would scrap silly things like 80 ton tanks, but this is Nazi Germany and its pretty obvious that optimal economic decisions will not be made.
We can expect more losses on the Eastern Front and the civilian death tolls, particularly that of Germans and to a lesser degree Soviets and Italians.
And German nukes are not appearing before 1960 or someone else's nuclear program, unless major changes are made to Nazi Germany's government.