Given that so many of the communities in the Celtic Fringe were illiterate until modern times, what they spoke in the distant past in any particular place is very little more than a good guess.
We do have a decent sampling of writing in the native languages from the Dark Ages and Middle Ages, and the selction is particularly rich coming from Ireland and Wales, which show the Goidelic/Brythonic split pretty nicely. There are also many, many examples of Roman inscriptions using local languages and names for deities, which strongly indicate that whatever local variations in languages may have existed, the general split was there. There's also the Irish Ogham stones, even the earliest examples of which are written in proto-Gaelic.
Of course, in the far distant past, we know they owned Greek scribes because that's the language in which their business and religious affairs were conducted.
That's a new one on me, as well. Are you talking about their pre-Roman, pre-literate past? I'm not aware of any examples of Greek writing which show Celtic business affairs - in fact the only example of pre-Roman examples of Celtic written records at all is the Coligny Calendar, which was definitely written in Gaulish rather than Greek. I will try to remember to ask Alexei Kondratiev about all this.
Most of our earliest materials concerning the comings and goings of the Scota, et al, are in Greek ~ these are not Greek records, but Scota records in Greek!
In an earlier time Sumerian played a similar part in the Middle East, and in a later time Latin played the same part throughout Europe and the Mediterranean.
BTW, the last report I read concerning what the Picts spoke suggested their language group had many Celtic words acquired from the Celtic tribes with whom they traded, but otherwise it stood alone (as does Basque).
BTW, it was still Alba up until the 9th century when the Scota and their Viking allies took over the place, so differentiation into Alba and Scotia slops over the Romano-Brittain period and can be confusing.