= Solution
Start with the countable model $M_0$. There are countably many finite tuples and formulas. For every pair $\bar a,\bar b\in M_i$ having the same type and every $c\in M_i$, use compactness to realize over $\bar b$ the transported type $\operatorname{tp}(c/\bar a)$. Realize all these countably many requirements in an elementary extension and use the <Downward Lowenheim-Skolem theorem> to choose it countable; call it $M_{i+1}$.
The elementary union $M_\omega=\bigcup_{i<\omega}M_i$ is countable. Any finite tuples and element in it occur at one stage, and their required matching element appears at the next. Thus $M_\omega$ is an aleph-zero-homogeneous <elementary extension> of $M_0$.
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