I've been researching the topic but so far all I've gotten is that people were angry about what the Enlightenment brought (e.g. the terror of the French Revolution) -- but to what extent does this apply in places like Britain, where the Revolution didn't occur?
From what I've read, spiritualism seems to have been a significant response in Britain to rationalism, but I'm struggling to find explanations to why it (and movements like it, in other places also) gained traction.
I can answer your question but frankly, I am a bit puzzled by the use of your terms.
Rationalism is a philosophical doctrine asserting that the place of reason is supreme and sufficient in understanding the world. This means that math, logic, arguments, etc. were sufficient to achieve Truth and that metaphysics, ethics and the like could achieve Truth through reasoning and other methods.
Spiritualism is the more general idea that there is an immaterial reality that cannot be perceived by the senses.
And my issue with your question is that... they do not contradict eachother. Descartes is definetely a good example of this; although a rationalist (even the rationalist by excellence, I'd say), his arguments in "Méditations Métaphysiques" lead to the postulation that there is a God and definetely point overall to a spiritualist philosophy.
An answer to rationalism would be empiricism but that does not seem to be in line with what you are providing me.
Let us just understand that, with the aforementioned definition of rationalism, its main opposing view (that mostly developped in Britain) is empiricism. Empiricism proposes that, contrary to rationalism, that knowledge is essentially derived from sensory experience as opposed to reason alone. Empiricism is definetely than not spiritualist.
I can go on but maybe you should clarifiy what you mean by spiritualism and rationalism first since, as you can see, rationalism and spiritualism are not exactly put in opposition or in reaction to.