British Embassy in Tonga
If you need assistance from the British High Commission in Fiji. This regarding matters related to Tonga, you can contact them directly for more information.
If you are in Tonga and you urgently need consular assistance (eg if you’ve been attacked, arrested or there has been a death), call +679 3304766 on the 24-hour helpline number.
You need to renew your British passport online and you can see here how to do that. If you have a BNO passport then you can see here how to renew that passport. Note that the British Embassy and British High Commissions no longer do this service.
British Birth Abroad:
Firstly you need to do this yourself as well. This needs to be done by post. Finally see here which documents need to be completed and where you will need to send the documents.