Well, thanks to you both, I really appreciate your time!
I am in a position to leave Cuzco on the 22nd, I get back from the Lares trek the day before.
I'm starting to think that perhaps a flight from Cuzco to La Paz could be an option. Not the cheapest one of course, but as I'm also keen to visit Lake Titicaca, and this won't be a trip I get to repeat in the foreseeable future, I'm seriously considering it.
My current, incomplete and possibly hopelessly optimistic, itinerary is...
- Sat May 23rd - Fly to La Paz - late afternoon/early evening in La Paz - catch Todo Turismo express bus to Uyuni
- Sun May 24th - Arrive in Uyuni early morning, start a tour late morning
Here's the hole. Either a two day or three day tour, I'm assuming a two day would be wisest. Then I'm aiming to get all the way back up to La Paz, and beyond to Lake Titicaca.
- Thurs May 28th - ...already at Lake Titicaca. Spend night in Copacabana.
- Fri May 29th - Travel back to La Paz
- Sat May 30th - Fly out of La Paz early morning.
As you can see, I'm hoping to avoid going through Chile at the moment. That's mainly to avoid missing Lake Titicaca, and also getting to Salar de Uyuni from the San Pedro side which I've heard is a brutal change in altitude.
What do you think - am I crazy?

Another option would be to shift everything a day earlier, and leave Cuzco on Fri 22nd May.
Thanks again for lending your expertise!
Matt