I have a staging copy of my site that is still on WP 5.1 and it shows the same issues.  Also I deleted LoginPress and the problem did not go away.

Could this be an .htaccess issue?

Hi Tucker and Richard,

I solved the problem (at least for me). It was not the login Press plugin, but simply switching off the 'WishList Member To Handle Retrieve Passwords:' option on the Wishlist Member pluging.

Thank you both for your effort.

So I installed a clean WP site 5.1.1 with Twenty Nineteen theme.  No plugins added but Frontend.  The only thing I have changed in the plugin config is the email text and the Senders Email fields.  Still sends me the default email text.

So the text stays the same, but the sender name and address change works fine?

No, the sender email name and address do not show different than the default.

Can you post the full email here please mate?

Here is the response from the WP site I just setup using 5.1.1 and I copied your default message and added a bunch of ZZZs and XXXs in front of the text so I would know if it worked (it didn't).  I also changed the username and email address, which does not show.

I am also having the same problem. It worked before I changed hosting companies and then it stopped. The difference in the servers is the version of php that they are using and that it is more strict with sql statements. If you could help me out, that would be great!