One minute changes are there as a training method to: "Enable you to make the chord changes at tempo in a song."
@pjby: Can you set a metronome at 60BPM strum one chord on each click for 4 beats/clicks, one bar, all down strums, and then get to another chord in time on the fifth click, the one of the next bar? If you can do that with and between the A, E, and D chords, you have effectively accomplished the intent of the exercise.
Many times issues like these are related to trying to go too fast too soon. Oddly enough to go fast you need to go slow. Slow is smooth and smooth is fast.
You might try something like this | A A A A | D D D D | E E E E | one down strum on each click of the metronome - or you might start only hitting the chord on the first click, letting 3 more click by and then hitting the new chord; so A 2 3 4 D 2 3 4 A 2 3 4 as slow as you need to go.
Also spend some time doing One Minute Changes slower. Forget about the timer. Just nice loose and relaxed, shift between the chords. Maybe even not shift. Grab the D chord, lift it off, grab it again, lift it off, grab it again, slow, mellow, relaxed. Repeat with the others and SLOWLY start to go faster.
Play some of the A D E songs. Slow them down so you can make the changes.
@set: I think people get way too hung up on the numbers, but I do think one minute changes it is a good tool. It is not the hammers fault if people decide to bash themselves in the head with it.
On the one hand, rarely if ever will you be required to change between one chord and another over and over for a minute. On the other hand you will need to be able to say, change between one chord and some other chord in tempo, at least that fast - and eventually way faster than the target speed of the exercise.