"Is it French, Great Britain, or German Enlightenment?" Enlightenment was a cosmopolitan project from day one (just like Freemasonry), as the French said "liberty, equality and fraternity". Kant defined it as you daring to use your own reason instead of relying on authority.
All Enlightenment thinkers believed that men by nature are rational (the Cartesian self-sufficient subject), that there is progress as a result of which tomorrow will be better than yesterday, and that positivist empirical science will not only make our lives better, but it will also solve the deepest mysteries of the universe. It was optimistic.
Postmodernists attacked it on all fronts. Freud ('the man is not a master in his own house'), Nietzsche, Lacan ('the subject is split') and Deleuze (schizo-analysis and disiring machines) dismantled the subject / ego / cogito.
Positivist empirical science was shown to be a system of domination over nature (for Heidegger) or over the non-privileged population (for Foucault). For Deleuze, science was a good thing, but he wanted a more 'horizontal' (he calls it rhizomatic) science, that does not oppress like the 'vertical' (he calls it arborescent or tree-like) science.
The notion of progress was challenged by Heidegger in his essay on technology, and in his rarely read text "Four Seminars" (this work is fascinating because it has the only mention of Wittgenstein that Heidegger made, and also discusses Marx at length).
Derrida challenged many ideas that lie at the foundation of Enlightenment, for him they exemplify "metaphysics of presence" that he opposed. For example, in "Of Grammatology" he provides an interesting reading of JJ Rousseau that uncovers inconsistencies in Rousseau's theory of the 'noble savage' and his views on masturbation.
"Also, what do successors of the Great Britain Enlightenment (i.e. in political philosophy, classical liberals, and in philosophy, I would say, empiricists) think of French postmodernists?" - The successors of British Enlightenment are analytic philosophers (which is arguable I admit, since Frege and Wittgenstein were from the continent). Russell said that ever since Leibniz there have been 2 very different ways of doing philosophy, the German / continental way and the ango-saxon / analytic way.
In the 21st century the predominant attitude is that of distrust, disregard and hate. The most illustrative example is the 1996 "Sokal's Affair". An American physicist Sokal submitted a nonsensical paper to a philosophical journal that published continental philosophy, and that paper got published with positive reviews. Sokal then said, you see, all of postmodernism is nonsense. Derrida replied with a terrific snarky essay called "Limited Inc".