I would certainly say instinct exists in the society Huxley has created, but it has been so carefully controlled and manipulated that people either don't realize it's there, or misread it as an annoyance or nuisance. Consider the Pregnancy Surrogates that women must occasionally take (designed to mimic the hormonal levels of menstruation) or the Violent Passion Surrogates that everyone must take. Mond even acknowledges that unless people are allowed an outpouring of emotion from time to time, they will essentially breakdown or somehow destroy themselves.
Some characters sense these instincts correctly and even follow them. John, of course, is the best example of this. He is outraged by the society he encounters in London, and routinely questions himself or whips himself to find a release. He also forces each person with whom he comes into contact to address their own sense of self. Other Alphas, such as Bernard and Helmholtz, also sense these fundamental feelings of doubt and unease. Because their caste is allowed more freedom, and their thought processes are not stunted by their creation, they are much more aware of their instincts at war with their conditioning. Finally, even Linda, raised in a society where "birth" and "parents" are ideas so shameful and disgusting they cannot even be conceived, raises John as best she knows how. No one would ever defend her as a great mother, but maternal instinct had to have played a role in her relationship with John, or he would have been an orphan from birth.