In a Riemannian manifold consider two compact smooth submanifolds $S$, $S^\prime$ that intersect transversely. It seems intuitively obvious that for a sufficiently small number $r$, the union of $r$-neighborhoods of $S$ and $S^\prime$ is a regular neighborhood of $S\cup S^\prime$ in some triangulation of the ambient manifold. Is this written somewhere?

By the way, a standard reference for regular neighborhoods is this paper by Marshall Cohen, published in Trans. Amer. Math. Soc. 136, 1969 189--229.