Without putting wordpress or any of those CMS's into the picture, the answer is clear.
If you upload a file to a server that is the exact same filename referenced by a link from any page on the web, then there is no need to update any of the links.
As much as I'm taking shortcuts here, this answer is easy to verify.
Simply create a simple HTML file with the following contents:
<a href="something.html">A link</a>
Save it as index.html or index.htm (depending on the first file your specific server looks for when a user requests the home page) then upload it to your document root folder on the server. To be on the safer side, upload the file as index.htm and make a copy of it and save it as index.html as well.
Next, create a simple HTML file with whatever contents you want and save it as something.html and upload that file to the exact same folder where you saved the index.htm or index.html file.
Access your domain (by going to www.whatever-your-domain-is.com in your browser) and you will see the following on a blank screen:
Click on it and you'll see the output of "something.html" after the browser processes the HTML.
Now on your local computer, change the contents of something.html and save it again, and upload it again to the same location, and if asked to overwrite the existing file in your editor, choose yes. If asked to overwrite the file on the server, choose yes. Now go in your browser, and clear the cache and then access your domain again. You should notice that when "A link" is clicked, you'll see your changes.
As you have noticed, no changes were made to index.htm or index.html.