The English tried outlawing several of the Celtic languages repeatedly. They never came to a great deal of success with it, except maybe inside of the Pale and in the Canadian Maritimes. Barring it as the official language of correspondance only meant that Nobles were barred from it as the vast majority were illiterate anyway and Monks and others would prefer French. Aside from that Shakespeare's Henry IV Part II testifies to the presence of Nobles who spoke exclusively Welsh as well. If you wandered around Northern Wales, speaking only English around 1500, you might as well have been speaking Russian. In Ireland, the worst damage to Irish language was done by a combination of the famine and then incompetant but well meaning governmental policies of the Irish Free State (due in part to increased globalization).
English has very minor Celtic influence that differentiates it from other Germanic languages, but far far more in the way of French influence. Most of what is celtic influenced is heavily disputed, one of the more significant and less controversial is the verb "do" which (initially) mimicked the Celtic habitual present. This is most visible in the old Hiberno-English construction of "He do be..." to indicate that someone is actively doing something. This is a unique quirk, sure, but hardly a defining corner stone of English. Especially when you consider that the use of "do" has shifted away from the more Celtic construction into something that's uniquely English.
Now, in terms of similarity; As an English speaker, I can squint my eyes and make out some Dutch and Frisian but a LOT of Scots and with my French, I can understand most of a technical article written in Spanish or Occitan. In the same way as learning Irish speaker, I can puzzle out a good bit of Welsh. It's not mutually comprehensible, no, but they are very closely related.
Historically and culturally the Celts had far more in common with eachother than the English. The English were a mixed culture from the start; originating and remaining close to West Germans but drawing substantial influence from the French (particularly amongst their nobility and educated classes), Scandinavians and yes, Celts. But that is hardly justification for merging the English into Celtic groups any more than chucking the French in with Germanic because of their Frankish heritage.
Now, to genetics; I'll have to say I'm very skeptical about trying to pin the culture of nations based on this matter. From my studies on it, the over all genetics of Europe have been fairly sedentary and likely predate the arrival of the IE peoples. Mixing has always occurred and these groups shift a little and there are points where it spikes, but it is language and culture which has us give narrative to this. The arrival of the Celts, expansion of the Roman Empire and the invasions of the Germans didn't really do much to fundamentally alter the genetic code of the people living in Gaul or Iberia. What it did change is language and customs. And I think it is clear that during the period from the 14th to 19th Centuries, the language and customs of the Celtic peoples of Wales, Scotland, Ireland, Brittany and Cornwall, shared a resemblance which did not exist amongst the English or French people who ruled them.
How different were English/Scotish/Highlander/Welsh/Irish cultures and their language at the time? More different than Bretton and French or Swiss and Prussian or Navaran and Portugese or Albanian and Romanian that they cant be in the same cultural group like these examples?
Here is a modern comparison of colour terms between the Celtic languages and English;
http://www.omniglot.com/language/celtic/connections/colours.htm#colours
The Scots language (common to the lowlands) reads like this;
https://sco.wikipedia.org/wiki/Scotland
I'd say the differences between this and English are roughly analogous as those between Scots Gaelic and Irish Gaelic. Reading Welsh or Breton would be rather like trying to read Dutch or Frisian. Indeed, Welsh and Breton are actually fairly close, except that the Bretons have adopted a few more French influenced spelling customs where the Welsh and Cornish are more insular.
I'm actually really annoyed by these moves to put Breton in the French Culture group and other mergers that didn't fly in EU3, so as I've said; putting the Celts and English into a "British" group would be inaccurate but it seems to match the general direction of things.
As to Culture? Yes, the Celts were profoundly different. Leaving aside language (along with that their traditional legends, stories and so forth) for a moment; their manner of dress, customs of rulership and inheritance, poetic and musical culture differed greatly and their classes system and laws were profoundly different until very late in the game due to British rule. Arguably, the United Irishmen (around 1800) represent the first major uprising which didn't draw on Gaelic nobility and law but were instead largely Protestant and influenced by French Revolutionary law. Brehon law, which long predates English occupation (and perhaps even the Saxon invasions) was in effect in Gaelic Ireland until the end of the 16th century with the Nine Years War and the Flight of the Earls. War was fought differently by Gaelic Gallowglass and Kern compared to the English Knights and Longbowmen. I really don't know what else you would need to differ between these groups; the only thing they had in common is that the both inhabited a group of islands off Europe and were (initially) all catholic and under Norman rule.
English really, REALLY shouldn't be in the Germanic group. Sure, English is a Germanic language, but that doesn't mean it's German. That's like arguing that Portugal should be in the Italian group because Portuguese is a Romance language and Roma is in Italy. It's just a terrible idea.
I agree with Germany at large, though I could see an argument for grouping the English with the Dutch, Flemish and Frisians if someone felt the urge to put them into a cultural group with some mainland Europeans. Though this would mean splitting Dutch from the German cultural group which would be a really weird idea for a lot of reasons, so don't take me as supporting it.
But right now, I feel as though the argument that Gaels and Welsh ought to be merged into a "British" group and Bretons belong under France is analogous to saying "Well, English were ruled by Normans who spoke French and wanted the French throne, so why not make English a part of the French Cultural group?" It's just lumping groups together based off of where the nation historically conquered which doesn't make a lot of sense at all. Nobility, Merchants and the Educated in England may have spoken French for several centuries, but very few would advocate making English a French subcultural group, because the vast majority of the inhabitants identified as distinctly English and brought their language to government once they rose to prominence. Why should the Irish be reduced to a British subgroup simply because they were ruled at times by English and Norman speaking Monarchs? Irish, Welsh and even the Cornish rose up at times under leaders speaking their own language and abiding by their own regional customs. There's no reason to believe that, if France's power waned and Brittany's rose, the Breton nobles would keep the French tongue rather than, like the Normans, adopting their regional language.
It's not as though France or England are often put into losing games for the sake of a couple poorer provinces at the edge of this historical territories having a Celtic culture. I really don't think the Big Blue Blob is going to collapse for the sake of a handful of Breton provinces, even under AI. Meanwhile, there are a lot of us who play Celt games where these handful of provinces make a huge difference and it is satisfying to fight and curb two of the most powerful nations both in the game and historically in order to forge an unlikely Empire. This move would take away one of the few advantages available to players trying for this and make it less fulfilling.