For an open source Android app for OpenStreetMap data, I like Organic Maps, and it normally works great with locally-cached maps. I've had better luck with it than with Google Maps or Apple Maps on phones.
(Though, I should mention that twice in the last year I've had Organic Maps become hopelessly confused about where I was, and where I should go. Both times, it had gotten a good GPS location, but then got confused while being out for an extended period of time, like maybe it was dead-reckoning only after that initial lock.)
(Though, I should mention that twice in the last year I've had Organic Maps become hopelessly confused about where I was, and where I should go. Both times, it had gotten a good GPS location, but then got confused while being out for an extended period of time, like maybe it was dead-reckoning only after that initial lock.)