How does changing my business email address affect Skype for Business?

Rather badly. Skype for Business is linked to a primary account and changing that can cause various issues. It would be better to keep your existing business email account for Skype and create a new Skype account for your new email address, and gradually phase that one in (i.e. booking new meetings/ conference calls on the new account, whilst still keeping the old account open for meetings/ conference calls that have already been set.)