Yes, it is the case that WordPress needs to be able to access it’s own URL via HTTP for those features to work, and also for wp-cron related tasks such as scheduling.
I also have customers with WordPress sites on Heart Internet servers with the same issue. According to Heart Internet, it is a “security enhancement” which they “cannot fix.” I’ve come to the conclusion that the only real solution is to move the sites elsewhere. Heart Internet are fully aware of the problem, refuse to fix it, yet continue to claim (to new customers) that their hosting is compatible with WordPress.