Spain and the United Kingdom's response
would only work if there was a single response jointly issued by both countries. It is the same as:
John and Mary's child John and Mary are the parents of a child.
John's and Mary's children John and Mary have children, but not together.
Using the possessive of the countries and a plural responses indicates different "parentage," but doesn't limit the number of responses to just one each unless it's clear from context:
Spain's and the United Kingdom's responses
Some find a double possessive like this rather ponderous, so you could use adjectives instead:
the Spanish and British responses
Here again, context would have to limit to one response each.
You could further lighten the title by using a colon and a subtitle:
"The Spanish and British Responses to [Whatever]: A Comparative Study"
We say a [comparative] study in a particular field:
A Comparative Study in the Aesthetics of Paul Valery and Edgar Allan Poe
A comparative study of x is simply about what is compared:
American & British Romanticism: A Comparative Study of Edgar Allan Poe & Other Romantic Poets