Hey there Jon-in-Chi,
Thanks for using Apple Support Communities!
Based on what you're experiencing, these steps can help.
"Things to check first
- Check the system status of iCloud Mail.
- Check that you turned on iCloud Mail:
- On your Mac, open the Mail app then choose Mail >Preferences. Click Accounts, then select your iCloud account in the sidebar. Under Account Information, make sure that the account is turned on and the status is online.
- On your iPhone, iPad, or iPod touch, go to Settings > [your name] > iCloud and check that Mail is turned on.
3.Make sure you that haven't exceeded your iCloud storage limit. If you exceed your iCloud storage limit, learn how to make more space available in iCloud.
4.Make sure that you're using the latest version of software on your device."
This information can be found in If iCloud Mail isn't working. You'd want to work these steps first then move through the rest of the guide.
You've also mentioned using a third-party service to send the emails. We'd also suggest checking their site to see if there is a setting they can suggest since you've narrowed this down to iCloud mail: Mailerlite. The way to narrow this down would be sending the email outside the service to see if it arrives quickly. By doing this it will resolve whether it's the service itself or using iCloud Mail.
Kind regards.