= Solution
For fixed $0\leq\alpha<1$, the phase <derivative> is $-1+\alpha\cos x\leq-(1-\alpha)<0$ on the integration interval. Its maximum is at $x=0$, with phase zero and <derivative> $-(1-\alpha)$. The <Laplace endpoint estimate>, or the substitution $x=t/[\lambda(1-\alpha)]$, gives
$$
\boxed{f(\lambda,\alpha)\sim\frac1{\lambda(1-\alpha)}.}
$$
This fixed-parameter formula is not uniform as $\alpha\uparrow1$; its diverging coefficient signals the <distinguished limit> derived in the parent solution.
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