Fear Not, THERE IS A SOLUTION

I recently accidentally deleted the wrong email account in Apple Mail.
I thought that all my emails were gone for good!

I got them back.
Here's how you can too:
-There a few different ways depending on how long it's been and other circumstances

First, before you do anything else (I made this mistake -waited too long).
Open your Time Machine (if you have it)
EVEN IF YOU HAVEN'T BACKED UP FOR A WHILE
-You will probably find that Time Machine backed up your data about 30mins and 1hr ago.
Note, to make it easier, you want to have Mail open when you open Time Machine so that it also goes into Mail right away. Then, highlight the emails you want and click 'restore' (or equivalent button).
Make sure you also restore any sent emails, drafts, etc that you want to keep as deleting a email account can often delete these also.

If it's been too long or you don't have Time Machine...
Second, log on to your webmail. -You may find that many emails have been left on there or there is a backup/archive. If you can't find any, contact your email service provider to see if and how far back they can go. I combined this method with an older backup of Time Machine to get almost all my emails back.
You may need to set up youremail account again in Mail to download them or have them forwarded/sent or downloaded* to another email address.
*Gmail (and perhaps others) has an option where your email account can 'Pop' (download) emails from another account. -This is a good option if you didn't want your account completely restored again anyway as was my case (I wanted the emails on another account but deleted them instead!)

Third, download data recovery software.