Short answer: You really can't.
Long answer: The most direct way (ie not taking a Potosi bus and connecting) is to take just two buses.
The first bus will be from Uyuni to La Paz. This can take 12-15 hours depending on the time of day, the bus company, your driver (some will pull over for a few hours to sleep!) and stops (some stop in El Alto to drop off ...
For most travellers, there's really only one logical way - the bus. The airport is 50km away in Juliaca, and generally you fly from Lima.
I did this trip by bus back in 2010 - the 'fast' option around the lake, and you can read my brief bit about it on my blog.
From the Article on Wikivoyage:
Daily buses depart to and from La Paz, Bolivia on two ...
By bus, the easiest is going to be a stop in La Paz, Bolivia.
You have two options from Puno. The direct - through the border and round the south side of Lake Titicaca to La Paz. It doesn't take that long - about 5 hours from memory, although I've heard it can take 8.
Option B is the ferry through to Copacobana. It's meant to be quite the experience, but ...
Just go to the Bus Station in downtown Laz Paz. There are several buses to Peru, most of them will stop in Puno. Just double-check when buying your ticket.
It may be cheaper to take domestic buses and walk across the border but buses around there are so cheap that it is hardly worth the trouble.