Saddle position is independent of handlebars, generally speaking... saddle position should be where you are best balanced fore-to-aft and with the appropriate leg reach to the pedals.

It could change though, depending on how upright you go. If you are going from an aero position to a 45degree position (or from 45 degree to 90) then that might be enough of a change that you will want to get a wider saddle (sit bones will sit differently depending on your angle). Obviously, if you have to change the saddle, you'll need to readjust position a bit.

But for the most part, you should be treating saddle position and handlebar position independently if the bike is a good fit to begin with. Still, as you move more upright, you may find yourself sitting back farther on the saddle than you were previously, necessitating a move. I doubt it, but it's not out of the question either. Just listen to your body.

I'm not a fitter though, so take it with a grain of salt.