In this article, we study the scalar one-dimensional nonlocal quasilinear problem of the form$$u_t=a(\Vert u_x\Vert^2)u_{xx}+\nu f(u) , $$with Dirichlet boundary conditions on the interval$[0,\pi]$, where$a: \mathbb{R}^+\to [m,M]\subset (0,+\infty)$ and$f: \mathbb{R}\to \mathbb{R}$ are continuous functions that satisfy suitable additional conditions. We give a complete characterization of the bifurcations and hyperbolicity for the corresponding equilibria. With respect to bifurcation, the existing result requires that the function$a(\cdot)$ be non-decreasing and shows that bifurcations are pitchfork supercritical bifurcations from zero. We extend these results to the case of a general smooth nonlocal diffusion function$a(\cdot)$ and show that bifurcations may be pitchfork or saddle-node, both subcritical or supercritical. Concerning hyperbolicity, we specifying necessary and sufficient conditions for its occurrence. We also explore some examples to exhibit the variety of possibilities, depending on the choice of the function$a(\cdot)$, that may occur as the parameter$\nu$ varies.