Randomly. An actual textual argument for Bush coming up through the standard young gentleman's way (AKA: not based on my, uh, totally thorough reading of two skinny little books and continued ignorance about anything): that bit in FC where Hornblower sees Bush brooding about what career he might have once he returns to see and consequently starts thinking himself about what he might be able to do for Bush and Brown upon their return to England.

He talks about how it's not unprecedented for men to rise from the lower decks when he's thinking about Brown, does a bit of justifying to himself and citing of precedent, but he does not when he's thinking about Bush. I mean, it's entirely possible that Bush never told him about coming up that way or that Horatio thinks it doesn't matter because Bush is already a Lieutenant, but yeah.