Ogham in Britain in general post-dates Roman Britannia (some may date from Irish incursions as the end of Empire approached) as do the Pictish inscriptions. The earliest of the Pictish stones are generally dated to the 6thC (though some say earlier) though these earliest stones have only picture carvings on them anyway! About 50% of all Pictish stones are thought to date from the last 150 years or so of Pictish power hence that would be centuries later. You are right in that there isn't a lot of evidence. It amounts to place names, king lists, earliest insciptions and of course the later surviving languages themselves. What evidence there is for Celtic languages in Roman Britannia points to P-Celtic languages though. That is accepted as fact by historians. In fact any evidence of language at all in Roman Britain (Latin aside) points to P-Celtic. There may indeed have been other indiginous languages but we don't know of them.

There are some undeciphered (personal names aside) inscriptions on 14 or so of the Pictish stones. Some say this could mean that apart from the main Pictish P-Celtic language there perhaps was also a minority language still in existence in Pictland which was not only not Celtic but was non-Indo European. Some suggest it may have been a dead language only used for religious purposes etc. As they are undeciphered no-one really knows.