For my credentials: I am a linguist-in-training, and though I have never studied gaelic languages but I did take a passing interest in them for several months and am roughly familiar with the general outline of them, if not any detailed scholarship. My main use here is mostly terminology and useful concepts you might not be aware of:
I was reading an article on the Cornwall road signs issue, and had some general questions about the languages of the Britain and Ireland:
1) Irish Gaelic and Scottish Gaelic are different - so which tribe spoke Irish Gaelic?
Irish Gaelic and Scottish Gaelic are roughly similar in the same way as British and American English are different- or perhaps Latin American Spanish and Castillian Spanish, if you are more familiar with those, as it's more apt in terms of intelligibility. They have different phrases and vocabulary in some parts, and different spelling (different spelling reforms, as the peculiarities of irish and Gaelic spelling leave a lot to be desired on the 'easy to learn' front, but it's eminently 100% readable with no errors once you learn the logic behind what all the 'silent' letters mean (some of which aren't silent but just things Anglophones aren't used to hearing, like the difference between what two hypothetical words 'aip' and 'ap' differ in regards to.))
The two aren't actually different languages, maybe, but that's because if you look too hard, nothing is a different language: Languages are a hierarchical tree of descent, something like animals radiating outwards and becoming extinct, except they're animals that can exchange bits and pieces of themselves (although not often the important bits- the grammar, usually just some cosmetic things like words).
When two languages split off, you find that at first they are comprehensible (and thus called 'dialects', although sometimes that word is used oddly like in Chinese, where the dialects aren't comprehensible), and indeed sometimes they merge back in (some theories about African American Vernacular English (which the unaware call Ebonics) believe that AAVE branched outwards and is now slowly falling back in, without ever having become truly incomprehensible to a native speaker of non-AAVE english.
Sometimes languages form a gradient, where North can talk to Central and Central to South, but Northerners and Southerners can't directly understand eachother (something like a quasi-realistic example of Austrians only being understood by Bavarians who can be understood by Thuringians who can- etc).
If Central Died out, in the example, North and South wouldn't be able to talk to a living intermediary, and thus would be two different languages. Except often people can understand eachother when they're not the same language, like French and Spanish, even though there's no dialects between French and Spanish (well, there are, but let's not get into how messy and diverse the romance language family was before the french literally beat non-Parisian languages out of their schoolchildren). Or the Swedish-Norwegian-Danish triad, with varying degrees of comprehension on each side. That's because this terminology isn't precise- like a colour map, people can cite when something is 'a different colour' (when, say, something is red vs orange or green vs blue), but can't really narrow down exactly where, because they blend into eachother.
So, to answer your question in a terribly roundabout fashion, I'm sorry, but language wasn't actually by tribe. It's not actually by anything, unless you say by person, to be exact. Every village has its own variety. But on the large scale, they all spoke Gaelic, to varying degrees. Whether someone from Meath could speak to someone from Ulster directly is something I don't know, but there were definitely intermediaries- for one, there are currently at least 5 modern dialects of Irish even in its much-reduced form in the Gaeltacht.
With that said, there is a specific kingdom we can identify with being explicitly Scottish. The Kingdom of Dal Riada had a territory that roughly spans modern Northern Ireland and the western half of the Highlands, including the Hebrides. The Dal Riadans were themselves displaced in some largely confusing population movements that I'm not quite knowledgeable about, leading to their assimilation into various other kingdoms in northern ireland. The Picts, for what it's worth, were partially replaced, and even before their assimilation into Dal Riada/Dal Riada's assimilation into Pictland (the two kind of form a confusing stew for which I have no satisfactory explanation in a simple manner of whom actually was the one to 'survive'), the Picts were using Gaelic as a courtly language. The long times of isolation and divergence explain the relatively minor grammatical differences and vocabulary.
2) Does the language that was spoken by the Picts fall under "Brythonic"?
To a degree, we have very little idea, given that we have literally 0 words in Pictish outside of names of places and people. However, Anglo-Saxon historians clearly distinguished Welsh, Pictish, and Gaelic as the three main languages of the British Isles. It almost certainly is a Brythonic language like Welsh (as opposed to the Goidelic languages like Gaelic and Manx), probably rather similar to one of the varying languages the Anglo-Saxons displaced from 'Hen Ogledd', the Welsh name for the northern Britanic kingdoms, and thus similarish enough to Welsh for Welsh writers to regard them, like all of the languages of that area, as just odd accents or distorted versions. (Depending upon how you look at it, you could also count Cumbrian as a language, but it appears to be substantially the same as Welsh from what little we know about it from half-remembered shepherding numeral systems (that is, not much of certainty). Then again, so does any Brythonic language, as the distinctions are fuzzy, especially with something as basic as numbers).)
The tree roughly goes:
oo Continental- Iberian (?), Gallic, etc. [not very well understood, which is an immense sadness to me]
ooo Goidelic- Irish Gaelic, Scottish Gaelic, Manx
ooo Brythonic- Welsh, (Cumbric?), Cornish, Breton, some other minor ones I've never heard of like "Ivernic", Pictish
Beyond this you get into dialects like Meath Irish, Ulster Irish, Northern Welsh, Southern Welsh, etc.
So assuming you mean Brythonic in this manner, then yes. If you mean 'the language of the post-Roman Britons', Welsh is a closer answer (although it was no more Welsh than Pictish or Cumbrian, but we've scant information on the various dialects continuums of southern England).
4) People talk about the rate at which the Brythonic languages are dying out - I know it's spoken, at least partially, in Wales (Welsh), and Brittany (Breton). However, obviously Welsh kingdoms were spread through many parts of Britain, so does the extinction rate refer to the Brythonic languages of places like Cornwall and Cumbria? If so, how many such languages are there - and how many have already gone extinct?
It depends upon how you define languages, and extinction. Cornish and Manx are dead but have artificially revived forms. With Manx, the time deceased was rather minimal, whereas it's been quite a bit of time since the last Cornish speaker died, yet there are now reasonably fluent Cornish speakers. With that said, they'll never be the same- their dialects are gone, and all their grammar and idiosyncracies are based entirely on what few resources we had, and thus those peoples' idiolects (personal speech patterns, like a tiny tiny dialect) rather than the language as itself as a whole. For a lesser degree, this has happened in Irish (with the loss of much of the diversity of the Irish language), and indeed any language (save maybe German and other language models which have not wiped out regional dialects but instead supplemented them).
As an aside, you also see serious aberrant english grammar intrusion (or whatever the replacing language is) in languages that are dying, dead, or revived, as the speakers are all generally native english speakers or native bilinguals, and thus import many calques, idioms, pieces of vocabulary, and even entire grammar structures into their language. As a particularly telling example, word order often swaps to Subject Verb Object like English, even in a Verb Subject Object language like the Goidelic languages, in the final stages of death. So Cornish and Manx, even if learned from direct audio recordings and collected knowledges of the final speakers, are probably irreparably changed. Which isn't to say worse- there's no wrong way to speak it. Just different, and probably negatively so in most eyes.
Breton is still alive, although shrinking. Scots is shrinking but the rate of decline is slowing with each decade. With that said, they still trend towards extinction, and their youth speaking rates are atrocious: it's more that more and more elderly Scottish speakers in the Hebrides are getting longer lives than anything useful done by the Scots. Irish is on the rebound, although in that awkward puppy stage where society is not truly bilingual nor do most speakers actually possess useful fluency. With that said, it's a unique example among its entire kind of a language brought back from literally the brink. I give great credit to the Irish government's cultural/radio/tv/music programs- people need something to do with a language to care about learning or keeping it- and also the more fortunate geographic placement and populations of the remaining Gaeltacht areas than the Scottish, as well as some quirks of history (Anti-Britishism -> Irish Nationalism -> Linguistic Preservationism) that make it a larger priority for politicians and speakers.
Cumbrian died due to Anglo-Saxons, although really old-timey English shepherds still used really bad loans of its number system for counting back when people still did stuff like sheep herding (sheepcounting bumfit for whatever the Cumbrian equivalent of Welsh Bymtheg was. That means Fifteen, by the way- the chance resemblance ofBym to Fif and Theg to teen (which isn't spurious, when you also take into account other words like Peddera (similar to both Four and Quarter)) are more evidences of the PIE underpinnings of the various european languages*, a triumph of 19th century scientific linguistics and a fascinating thing we take for granted nowadays, but which you haven't asked about so I will not diverge too much on). British itself, whatever number of languages that was, is dead. Welsh is undergoing some... okay... but less than stellar revitalisation attempts, but isn't dead yet. Everything else is dead. Pictish was killed by the Dal Riadans, although it made Scottish Gaelic in response.
*Except Hungarian, Finnish, Saami, and Basque, which are Finno-Ugric for the first three and a completely unknown thing which has no surviving relatives (except the ancient Vasconian, which was spoken by people in the exact same spot called the exact same thing and had similar naming elements and roots, which tends to imply that Vasconian is just a fancy way of describing "Old Basque" [pretty much a well accepted hypothesis by now]). Some people will tell you they have some crazy relation of Basque to something else, but they're damnable liars and like Sumerian it's completely a special snowflake because apparently far too many of their relatives died out and we're stuck like people who've only seen modern animals trying to trace what the possible ancestry of future descendants of those spike-walker things from the Cambrian Explosure are with fossils that utterly depended upon people having writing and writing it down (meaning a hard wall at a rough time of 10,000 BC, and a soft wall at anything before the printing press).
But I digress. Final total? Approximately... 4 Alive (Irish Scottish Welsh Breton), 3 Dead (Cumbrian, Non-Welsh Brythonic (we'll hope they all had one language), Pictish), 2 potentially twitching frankenstein monsters (Manx and Cornish). With all the caveats about the impreciseness of records and how you define a language.
Also, a final answer to a question that, if you've considered a map, is bugging you- Brythonic languages are on Britain, Goidelic on the Isles, and Continental on the Continent. Scottish is explained by the Dal Riadan Gaels invading Scotland at one point. Why is Breton, an Insular language, on the continent rather than Britain? Post-Anglo-Saxon invasion, or contemporaneous to it, British refugees apparantly fled to Brittany, which is why it's called Brittany ('little britain'). Thus why, annoyingly, exactly 1 Goidelic language is on Britain and 1 Brythonic language isn't.