Did the first wave of Celts arrive in Britain before La Tene? If these were Q-Celtic speaking, how does it explain an overwhelmingly P-Celtic population in Britain? Only Ireland spoke the Q form.
Here's what I say on the matter in The Peopling of Europe
: Beaker Folk to Celts and Italics
So what language did the Bell Beaker folk bring to the British Isles? Why were two types of Celtic spoken there by the time we have any records? Gaelic seems the older form. We can picture the first Beaker arrivals speaking an archaic form of Celtic that evolved over the millennia into the Irish, Manx and Scottish Gaelic that we know today. By contrast the Brittonic (or Brythonic) language of Britain was closely related to Gaulish, spoken across the Channel by the Roman period. That suggests that Britain received more or heavier waves of Celtic migration than did Ireland, continuing into the Iron Age. This fits the archaeological picture.
I elaborate in Celtic tribes of the British Isles
Other than archaeological evidence, the biggest clue to migration in prehistory is linguistic. By the time the British Isles enters true history i.e. its inhabitants were producing literature of their own, five languages were spoken within the islands, as Bede recorded. Latin was the language of the Church. The other languages were native to its mixture of peoples: English, British, Irish and Pictish.  English - "the language of the Angles" - had arrived with the Angles and Saxons. Norse was to arrive with the Vikings.
British, Irish and Pictish were all Celtic languages. The Irish spoke Gaelic, acknowledged by linguists to be the more archaic form of Insular Celtic. So we can picture this language gradually developing from an early form of Celtic spoken in the Bronze Age throughout the British Isles, when there was a good deal of contact between Ireland and Britain. It survived into historic times as Irish, Scottish and Manx Gaelic. Bede assumed that Gaelic arrived in Scotland with Irish people forming the Kingdom of Dál Riata just a few centuries before his time. His view was unquestioned until recently, when a lack of place-name or archaeological evidence for it was argued. Ewan Campbell suggested that Gaelic simply remained in the Highlands and Islands of Scotland from early times, untouched by the linguistic developments south and east of the Grampians. The most telling evidence against this is linguistic. Scottish, Irish and Manx Gaelic all descend from a common ancestor (Primitive Irish), which is attested in ogam inscriptions of the 5th and 6th centuries. Calculations of the date at which it split into dialects vary from 700 to 900 AD.. Furthermore there is archaeological evidence of long-distance contact in the Iron Age between the Celts of what is now Brittany and SE England on the one hand and Orkney, Shetland and the Western Isles on the other. (See the discussion of brochs below.) So the Highlands and Islands were not as isolated from southern developments as one might imagine.
British, known to linguists as Brythonic, and Pictish fit into a family of Celtic languages in which the "kw" sound of Indo-European had shifted to a "p" sound. P-Celtic was spoken in Gaul by Roman times, so we can deduce that Gaulish arrived in Britain with Iron-Age migrants and gradually developed into different dialects in the north and south of Britain, which eventually separated into distinct languages. The surviving form of it is Welsh. Another form - Cornish - survived in Cornwall into Tudor times.
Cumbric was the form spoken in what is now northern England and Lowland Scotland as far north as Dumbarton during the Early Middle Ages. It was closely related to Welsh. As Anglian settlements advanced, it was replaced by English and its Scottish variant - Lowland Scots. Pictish was the language of the south- eastern Highlands. It was similar to Cumbric. It had vanished by 1140, partly perhaps because the language of the Scottish court had changed to Gaelic when the Gaelic and Pictish kingdoms were united in the mid-9th century into the Kingdom of Alba, north of the Forth-Clyde isthmus. In the 10th and 11th centuries Alba expanded to include the Kingdom of Strathclyde, formerly part of Cumbria, in the southwest and Lothian, formerly part of Anglian Bernicia, in the south-east, and became known as the Kingdom of Scotland.
1. Bede, The Ecclesiastical History of the British People
, ed. J. McClure and R. Collins (1994), chapter 1.
2. E. Campbell, Were the Scots Irish?, Antiquity
, vol. 75 (2001), pp. 285–292.
3. J.T. Koch (ed.), Celtic Culture: a historical encyclopedia
(2006), p. 831; Václav Blažek, On the position of Gaulish within Celtic from the point of view of glottochronology, Indogermanische Forschungen
, vol. 114 (2009).
It is much easier to follow all this if you read the actual page - Celtic tribes of the British Isles
- , which has many maps.