Bumgenius are great, they dry overnight on an airer. The inserts they come with aren't great for heavy wetters though so would need boosting with a little hemp or bamboo. I find that hemp dries slightly quicker than bamboo.
In terms of fitteds, our Dunk 'n Fluff dry the quickest, but still a couple of days indoors in winter.
We've found the P'tit technic is almost dry when it comes out of the machine. Certainly the speediest drying, best fitting shaped 2 parter nappy we've come across. But pockets with microfibre inserts would probably be your nest bet for speedy trying. The Tirsokas pocket is particularly quick to dry - well certainly almost dry out of the machine!
Weehugger shells dry pretty quickly, and if you use inserts like the Flip stay dry they also dry really quickly. You could get a ton of flip stay drys and a few bamboo boosters if you LO is a massive wee'er?
The Lollipop microfleece was great with a newborn as it was dry pretty much right out of the machine. Fastest here is the Pop-in Dream Dri- only takes a few hours, and the bumGenius is usually dry overnight. Especially in winter when hanging near a radiator.