There isn't really any way to know this. Any writings that would shed light on this matter tend to be from the perspective of people within Rome itself, or sometimes from the Rhine legions. We don't really know from a practical standpoint how channels of civilian communication would work.

That being said, to speculate somewhat the first thing an emperor needed to do upon securing power would be ensuring the loyalty of the legions, and as Britain controlled two legions it would be an important point of communication. ORBIS gives a time of 9.1 days as the fastest route from Rome to London (I put in summer because Philip came to power during campaigning season) using horse relays and military routes, but to be more reasonable, we can assume the governor in London would receive news within two weeks. I doubt he would announce it to the townsfolk immediately upon hearing the news, but it wouldn't be too long after, so I suspect within the provincial capital of London news would trickle down within a month.

But this really only goes for London itself, for the rest of the province we don't even have the bare bones of information I am hanging my speculation on here. The second largest town in Britain was Cirencester, which would only be a few days of good riding from London. But how important would it be in practical terms that this information trickles out through the province? Emperors had been thick on the ground in the past decade, and perhaps to a private citizen in Britain the exact identity of the ruler was not seen as so important. In which case we could imagine the news spreading through the standard social networks of gossip and conversation. Information would only be crystal clear if a set of coins was distributed through Britain, but coin issues were often somewhat ad hoc and far from uniform, and I am uncertain whether Philip issued a series in Britain.