It depends on a few things. When I recently switched ISP and from dial up to "broad band" (or what passes for broadband where I live) I kept my dial up account with the old ISP until after the new service came on line so there was no period of time that I did not have internet. More recently my ISP provider changed my plan from one kind of broadband to another and there was supposed to be up three hours when I might be without broadband or a phone (as my ISP also provides my home phone line), but if that happened, I never noticed it.
When I swapped ISPs it took approximately 12 days or so before the new services came online, but this was over the period just after Christmas and over into the New Year, so there were a number of public holidays, and according to my ISP, greater delays than would usually occur.
Your new ISP provider should be able to tell you when their service will commence and if it begins before your current ISP service lapses, then you probably will experience minimal downtime, if any at all. If your email was provided by your old ISP and you are not retaining it, be sure to contact everyone in your address book and let them know your new email address (I avoided this by retaining my old email and paying a small monthly fee for it with my old ISP....the notion of changing my email address makes me a bit anxious).