It's going to be difficult to be definitive, but based on the context, the "Dual Alliance" was more likely the Russo-French Alliance of 1894 than any understanding involving the UK.
I do not think that exaggerates the easy complacency of, let us say, 1895, forty-five years ago. It was a complacency that lasted with most of us up to 1914. In 1914 hardly anyone in Europe or America below the age of fifty had seen anything of war in his own country.
The world before 1900 seemed to be drifting steadily towards a tacit but practical unification.
H.G.Wells, The New World Order
The Triple Alliance between Germany, Austria-Hungary, and Italy was formed in 1882. Italy left the alliance when World War I started.
The Triple Entente between the United Kingdom, France, and Russia was the coalescing of three bilateral agreements:
Your quote, and the following text which I have quoted, is describing the zeitgeist Wells perceived of the time leading up to World War I. However, this section of the novel invariably puts the zeit firmly in the late 19th Century. So the only part of the Triple Entente in place at the time was the one between Russia and France.
Of course, Wells is describing the complacency he felt at the time, from 45 years later. It is possible Wells wanted to underscore this complacency by being sloppy, and ignored the understandings involving France or Russia. Or ignored Italy.
As a footnote, prior to the Triple Alliance, there was an 1879 "Dual Alliance" between Germany and Austria-Hungary. But if this were Wells's Dual Alliance it would be unclear which Triple Alliance he meant.
