Nawab was how Indian princes/ kings were addressed. Shauk is the Urdu word for interests/ hobbies.
Nawabi shauk therefore literally means, a kings hobbies. In the days when nawabs and kings ruled kingdoms in the Indian subcontinent, only the rich and powerful (read nawabs) had access and the right to indulge in homosexual acts. Hence, acts male homosexuality came to be known as nawabi shauk and a gay man is described as someone having nawabi shauk.
Prince Manavendra Gohil, true to his name, has nawabi shauk (This sounds much better in Hindi, honestly).