To some extent both define a kind of fuzzy neighbourhood for each point, and both kind of induce a topology on the points. Both also seem to end up with some kind of weighted graph.

That said the UMAP approach has some very deep mathematical foundations, so it could take you a while to work that one out for a LSH approach.

That said you could also just use a few random projections as an initial step and use UMAP from there. The random projections should also give you a lower bound for the distance, which can be useful in finding the k nearest neighbours.