British Embassy in Tuvalu
If you have lost your passport while in Tuvalu then you will need to contact the High Commission in Fiji for an emergency travel document. If however you wish to renew your passport then you need to do this online which you can see here. If you have a BNO passport then you need to see the process and application process here. You can also track your passport online which you can read about here.
UK Visa Application:
The British High Commission does not perform this service any longer and you will need to start the process online with the UK Visa online process. This is where the process starts. Note that you will need to have a TB test to enter the UK long term and there is a listing of where to get this done.
British Birth Abroad:
The High Commission does not do this any longer and you will have to do this by post. You can find the needed documents online here.