Once you make the updates with your Registrar it could be a fast as 20 minutes, before your Domain is being pointed to the new Hoster. All new eMails will then be going there (situation normal) and if you still have access to your old Hosting account and presuming your CP is Cpanel, FTP in and download everything in the 'mail' directory -- one level above 'public_html'.
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If you will not change DNS, your site will function 100% normally during registrar transfer. No emails and no visitors will be lost.
Let's make that 99% instead, assuming whoever's hosting the site and/or email
doesn't experience any problems while the domain registration's transferring.
Tornado54, it depends on how soon the transfer is communicated and approved
with the gaining and losing registrar. IIRC (I think it was Stan's), it took at least
27-28 hours, the longest can take 7-10 days.