In my experience the emails get delivered almost instantaneously. If it takes longer in your case it’s worth considering changing the signup flow such that the email has more time to reach the user’s inbox. (e.g. add a step in between asking for their email and having them confirm it).
I’m pretty sure Gmail automatically shows the email message the moment it comes in without needing to reload the page. I imagine other email providers do the same. In my experience it all happens fast enough, but it’s worth testing.