Chesterton always is talking about German philosophers he despises mainly Nietzsche, even if he doesn't name them. He also routinely takes on his British contemporaries (and friends) spewing these same philosophies.
But you're right, he's not talking about George Bernard Shaw or H.G. Wells here. He's talking about what he saw in British Churches in the early 20th century.
I like to go back to what he says in Orthodoxy about Original Sin being the only provable doctrine. I think what he's saying is that there's a good many Christians who don't think they need 'a washing. Keep in mind the various ways Protestants and Catholics deal with post-baptismal sin and you'll see where I'm going. The doctrine of original sin means that we've got a tendency to follow our passions and they will lead us astray. Not all Christians believe this and a good many believe that if something feels good, that it's OK.
But, to your point, who were these people
Whether or no man could be washed in miraculous waters, there was no doubt at any rate that he wanted washing. But certain religious leaders in London, not mere materialists, have begun in our day not to deny the highly disputable water, but to deny the indisputable dirt. Certain new theologians dispute original sin, which is the only part of Christian theology which can really be proved. Some followers of the Reverend R.J.Campbell, in their almost too fastidious spirituality, admit divine sinlessness, which they cannot see even in their dreams. But they essentially deny human sin, which they can see in the street. The strongest saints and the strongest sceptics alike took positive evil as the starting-point of their argument. If it be true (as it certainly is) that a man can feel exquisite happiness in skinning a cat, then the religious philosopher can only draw one of two deductions. He must either deny the existence of God, as all atheists do; or he must deny the present union between God and man, as all Christians do. The new theologians seem to think it a highly rationalistic solution to deny the cat.
So, who is R.J. Campbell? Well, he's one of the progenitors of the New Theology Movement. And what is the New Theology Movement?
The New Theology is a false "religious" system that became popular among Catholics in Europe from the 1920s onward. Because it was recognized as resurgent modernism, it was kept under a lid by the Vatican and was condemned by Pope Pius XII in Humani Generis. Its primary founders were Maurice Blondel, Father Henri de Lubac and Hans Urs von Balthasar. It enjoys an undeserved popularity today.
A Short Catechism on the New Theology
I love short Catechisms that get to the point. So, there was a movement somewhere within liberal Anglican and liberal Catholic circles to apply modernism to theology, which fortunately was quashed rather handily in , but unfortunately oozed out over the 20th century and beyond.
A theology based on Evolution would sincerely deny original sin, since it denies our first parents' existence. That's one thing Catholics are not allowed to believe in regards to evolution
Such fictitious tenets of evolution which repudiate all that is absolute, firm and immutable, have paved the way for the new erroneous philosophy which, rivaling idealism, immanentism and pragmatism, has assumed the name of existentialism, since it concerns itself only with existence of individual things and neglects all consideration of their immutable essences.
Pope Pius XII - Humani Generis
So, thanks for asking the question (it was fun to track down all those references, I usually just assume I'll never know who he's talking about) and keep reading Chesterton!