My question is simple: Could you please explain (unfortunately, it means perform an step-by-step calculation) why the components of double covariant is given by $(1)$?
$$ (\nabla_{u}\nabla_{v}w)^{a} = u^{c}v^{b}\nabla_{c}\nabla_{b}w^{a}+(u^{c}\nabla_{c}v^{b})\nabla_{b}w^{a} \tag{1}$$
It's a direct computation using the Leibniz rule: $$\nabla_u\nabla_vw = u^b\nabla_b(v^c\nabla_cw) \stackrel{(\ast)}{=} u^bv^c\nabla_b\nabla_cw + (u^b\nabla_b v^c)\nabla_cw.$$