Pointing up or down isn't going to make any difference (I'm well endowed so I have to point up). And really it may not be the fault of the diaper anyways, unless you were using a really cheap brand, used it and took it off, then put it back on again later.
Generally though, any decent or premium diaper is going to lock away the wetness and keep you dry enough not to put you at risk. It also takes something like 8+ hours for bacteria to take root in a warm and wet diaper. The odds of getting a UTI before that are always going to be there regardless, but at the same time those odds are going to be incredibly close to zero in those first 8 hours.
Ps, I'm also a little confused. Your tag name says you're a DL, but you say you told the doctor you have a diaper fetish. Obviously a DL by definition doesn't have a sexual diaper fetish. Could you please clarify what you meant, or why you said that to your doc?