When you change your name servers it might affect the email delivery for around 24 hours while the dns updates.
So if you're changing web hosts make sure you still leave your old hosting account online around 3 days before you close it so that the old web hosting account can catch any incoming emails and incoming website traffic.
So leave your blog or website on the old host for a few days after changing web hosts so you won't lose out on website traffic and your website won't be down for some people.