Hopefully his British passport will have been stamped with PP presented or something similar.He is a dual citizen so hopefully he has a copy of his oath of allegiance taken when he reacquired Filipino citizenship under the scheme. If he can present that then I do not see any real problem. He has his British passport so has correct documentation for entry to his destination (UK)
The easiest solution if these are the facts is to apply for renewal of his Filipino passport. Just apply for appointment on DFA website, complete the requirements online and DFA will email a reference number to print and an appointment date and time.
Theoretically he should have no problems leaving with his British passport if he was stamped in correctly. The problem at Immigration is the high turnover of BI personnel due to the Tier2 trafficking issue. There are some very inexperienced officers working the counters.

I have a Filipino friend in London who visits every year.He let his British passport expire and travels on his Filipino passport. He carries a copy of his British naturalisation certificate and has no problems at check in or immigration.....he just spends 2 hours in line at Heathrow with the Nigerians,Jamaicans and Somalians before being admitted.