I start my gymnasts out with 30 second holds rather than pushing 60 seconds of ugly handstands and falling all over the place. For pre-team and compulsory level girls, they seem to pick up pretty quickly and advance to the one minute mark if they are worked consistently. They also have some sense of body positions, shapes, and awareness which helps.
With the high schoolers I'm currently working with, 30 second handstands in the proper shape is incredibly difficult (they were taught to do them backs to the wall with all kinds of crazy angles). So with them I'll do 30 second holds 4x with a partner holding their feet- this way the handstand-er and the partner can get a better sense of correct body position. At least that's how it works in my mind, rarely does it actually play out that way... I love to eventually work them up to 45-60 second holds, but am taking it very slowly. So don't feel bad if you need to start out small- good handstands require lots of practice and patience. Also, I prefer my gymnasts to do their wall handstands stomach against the wall to encourage better positions. There is still room for error, but not as much as with back to the wall handstands.