That's what I have to do - at the very least, I have to unwind all the strings to completely loose so I can stick my hand through the hole. I hate string through bridges for this reason. I'd rather have bridge pins if they're through body strung.If you're replacing a uke with a string through bridge, you may want to do them all together so that it's easier to turn the uke upside down to fish out the strings all at once. And then fish out the ends of the new strings to tie the knots or add the beads.
Restring one at a time, or remove all old strings and then replace?