Given a compact planar region $A$, let $τ_A$ be the (random) time it takes for the Johnson-Mehl tessellation of $A$ to be complete, i.e. the time it takes for $A$ to be fully covered by a spatial birth-growth process in $A$ with seeds arriving as a unit-intensity Poisson point process in $A \times [0,\infty)$, where upon arrival each seed grows at unit rate in all directions. We show that if $\partial A$ is smooth or polygonal then $\Pr [ πτ_{sA}^3 - 6 \log s - 4 \log \log s \leq x]$ tends to $\exp(- (\frac{81}{4π})^{1/3} |A|e^{-x/3} -(\frac{9}{2π^2})^{1/3} |\partial A| e^{-x/6})$ in the large-$s$ limit; the second term in the exponent is due to boundary effects, the importance of which was not recognized in earlier work on this model. We present similar results in higher dimensions (where boundary effects dominate). These results are derived using new results on the asymptotic probability of covering $A$ with a high-intensity spherical Poisson Boolean model restricted to $A$ with grains having iid small random radii, which generalize recent work of the first author that dealt only with grains of deterministic radius.