I doubt your cousin can apply directly for a British passport.
Section 12 of the British Nationality Act 1948 lists those British subjects that became CUKCs at commencement of the Act in 1949. Subsection 8 of that section listed who were CUKCs by descent only, who could not pass on their British citizenship to their children automatically. Your uncle was a CUKC and later a British citizen under Subsection 2.
He could not pass on his British citizenship to his children automatically. As your cousin was born after 1983, he could have registered her as a minor if either he had lived in the UK for three years before her birth or if he had moved to the UK and lived with her in the UK for three years before she turned 18.
As she is past 18 now anyway, there is not much she can do to acquire British citizenship based on her descent. She is of course eligible for an Ancestry visa.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.