I am afraid that Stephen has nothing whatever to do with The Ukulele Orcherstra Of Great Britain.
To answer your question, though. It really is just practice.
Try this.
1. Learn each chord separately. Make sure that all the strings are sounding clearly. If not, make sure that each finger is close to, but not on the fret.
2. Practise very slow the chord changes. Stop if necessary to change chords. Make sure you get the chords right.
3. After a while, try getting the changes smoothly, even if you fluff the chord sometimes.
In other words, sometimes concentrate on getting the chord right and sometimes getting the timing right. Don't do it at speed too soon.
Hope this helps.