The usual form of local limit theorem is extended to an arbitrary supercritical Galton-Watson process with arbitrary initial distribution. The existence of a continuous density on $(0, \infty)$ for the limit random variable $W$, in the process initiated by a single ancestor, follows from the derivation.
"The Local Limit Theorem for the Galton-Watson Process." Ann. Probab. 4 (3) 490 - 496, June, 1976. https://doi.org/10.1214/aop/1176996100