How to Change Chords Without Stopping the Strum
A chord change is fast and clean on its own, but as soon as a strumming pattern is added, the hand stalls or loses the beat right at the change. That's a separate skill from the chord change itself, and it needs to be trained separately.
Why the hand stalls on its own
While your attention is on forming the new chord, the strumming hand goes without a command for a split second - and stops, because up to that point all your attention was on it. This isn't a bad habit on its own, it's a natural result of the brain being able to hold only one complex task in focus at a time, until both become automatic.
Train each hand separately first
Before combining the chord change with the strum, get both parts automatic on their own: the chord change with no strumming at all (the one-minute-changes method), and the strumming pattern itself on a single chord, with no changes. Only once neither part needs conscious attention does it make sense to combine them - otherwise you're training the stumble, not the skill.
The 'skipped hit' trick instead of a full stop
The strumming hand shouldn't freeze at the moment of the change - it's better to deliberately skip one hit on the strings (the hand keeps moving through the air without touching the strings) than to stop the motion entirely. A skipped hit is almost inaudible and keeps the hand in rhythm, while a full stop is what actually reads as a rhythmic break.
Use the strumming pattern itself to hide the change
Most strumming patterns have a natural short gap - for instance, right before beat one of the next bar. If you move the chord change to land exactly there instead of mid-bar, the transition sounds far smoother: the listener hears a change in chord color, not a break in rhythm.
Progress: from one pair to a whole song
Drill the 'change plus strum' combination on one problem chord pair at a very slow metronome tempo, raising the speed only after several clean runs in a row (the same principle as in the metronome article) - and only then carry that solid pair into an entire real song.
Try it in practice
Open chord trainer