If you only have to change one string, the guitar'll go out of tune as Mr Ed said, but once you have the new string in place the others should be back to normal(ish). And the Sperzels will make it easy.
If you're changing the whole set, unless you want to clean the fingerboard or something, change them one at a time. That way each new string will be pretty much properly in tune as you go along.
If you take all the old strings off at once, say you tune the low E first, then it's fighting the spring tension all on its own. Then you fit the A, tune that up and the E's gone flat. And so it goes, you have to keep tuning over and over to "zoom in" on the correct pitch.
And that's why I don't like trems

. At least you haven't got a Floyd, they're a
real pain.