Since the 4 hands are all equal multiples, you should only need two shafts and a bunch of 16:1 reductions, like couples of 10 and 160 teeth. Second hand 10-tooth gear drives a stub 160, coupled to a 10, which drives 160 on the 4-minute hand, coupled to a 10... In my head, gear trains get more complicated when they need more shafts, because making sure all those different shafts are aligned and non-interfering is hard. Just stacking more 10/160 gears on longer shafts is pretty straightforward. Your clock would be deep, but not necessarily large.
If you have a 3D printer, you can download models of gears from places like https://mcmaster.com and print them out. Or modify them and print them. A 3D printed clock isn't likely to last until 2038, but it's great for prototyping.
I think we really need to know why sports are segregated.
If it's because males have physical advantages over females, then you really want to separate by some kind of size or strength criterion. Weight classes, like wrestling or boxing.
If its because we don't want children titillated by seeing the opposite sex change clothes in the locker room, then you need to come up with some way to address people with same-sex attraction.
If it's just because that's the way sports were when you were a kid, then let the two transgender kids play with the 10,000 cisgender kids.
FINA already bars trans-women who went through male puberty from competing as women, and that seems like a pretty fair compromise. At least until you pile on states trying to ban medical care for transgender youth - i.e. puberty blockers - or other difficulties many trans-youth have in obtaining such care. My understanding is that HRT after puberty doesn't come anywhere close to the biological effects of actual puberty, in terms of strength, size, and speed.