After a little bit of hunting around online, I found this site which told me what the Data APN should be.
Coming from オーストラリア, my Data APN would normally have been yesinternet as I am on the Optus network, however it was blank. I assume this is because my phone determined they were not the correct settings for the Solomon Islands and removed them. In the Solomon Islands though on the Our Telekom / Breeze network, it must be set to internet.
My phone did not receive these settings automatically unfortunately, but after that was entered data worked like a breeze. Pardon the pun.
On an iPhone you can find your mobile data settings by navigating to:
Settings > Mobile > Mobile Data Network
You can then set your data and personal hotspot APN’s which will normally be the same.