Everybody needs proper papers to travel, whatever the age is.
The baby needs a passport to travel. If the Father in English and recognize the baby (he signed the Filipino Birth Certificate), then he should go to the UK Embassy to declare the baby (if I remember well, the UK Citizen declaring a baby birth must be present) and request a British passport. Having a British passport for the baby may take some time, but is the best thing to avoid any problem for the travel (in case of Filipino passport, the baby will need a visa and surely a DSWD clearence to be free of travel).
Another thing, when buying the ticket for the travel they should request a baby bassinet (have to be done in travel agency, as it's not available online). As only few are available on each flight, they should also arrive in more in advance at check-in to have a confirmed baby bassinet for the flight (nota: in general it's available only on long-haul flights and not domestic connecting flights).