score:10
The Bureau of Immigration has now released the below statement. After speaking with them, they have confirmed that once the restrictions (Enhanced Community Quarantine) are lifted and the offices are open as normal again, the visa can be extended with no additional charge. That is, provided the request to extend it is received within 30 days of the restrictions being lifted.
Thanks to @lambshaanxy for pointing me in the right direction
Update 19th May 2020
Well, I'm still here. The most recent update is that the relevant paperwork will be issued at the airport when leaving, with no need to go to a government office to obtain the relevant extension/clearance before departing. I intend to get hold of someone somewhere to confirm the process and that this is the case, but at the moment that's all the information I have.
Update 25th June 2020
I'm still here... The previous information was incorrect (or rather, it is now incorrect) and after calling the Bureau of Immigration I was advised I need to extend my visa before end of June or face a penalty. All standard penalty fees are being waived until end of June but you still have to pay per month overstayed. My visa (as a UK passport holder) expired on 8th April and it cost 11,500PHP to renew it until 7th August. I can renew the tourist visa every 2 months (cost 3000PHP each time) for a maximum of 2 years. So at least now I am legal again, and can renew again August if necessary by visiting the local Bureau of Immigration office.
Upvote:10
The British Embassy or UK Foreign Office can't help you here, this is an internal matter in the Philippines. Fortunately, Tuguegarao is the capital of the province of Cagayan, so it has a Bureau of Immigration office (map). Given the circumstances, they should be able to arrange a visa extension or at least provide advice on what to do.
IMHO your best bet is indeed likely to hunker down and wait for the lockdown to end. This is currently supposed to happen on April 14th, but obviously this may change.