The term WASP has very little to do with actual descent from people who lived within Anglo Saxon culture in Britain in the centuries that it was the primary culture in England (AD 450 - AD 1100). As such WASP is a modern term used politically to single out, often in a derogatory way, wealthy people who are largely descended from British migrants.

The British people that migrated to the New World in time to establish the 'WASP' social group or stereotype in the U.S were a genetic mix of layered migration events extending from the relatively recent Huguenot refugee's fleeing France in the early Eighteenth Century. Back through millennia of invasions and migrations to one of the earliest major migrations we can see in the archaeological record, the Late Neolithic and Bronze Age Beaker Culture who appear to have migrated or at least injected new culture and ideas about 2500 BCE. With the earliest migration occurring soon after the last Ice Age around 12000 BCE.

As such the map of genetic backgrounds in Britain reflects these invasions but is far more of a blend than our historical sources had led us to believe. As this excellent website displays in far better detail than I can:
http://www.oum.ox.ac.uk/settlers/

If you were to read some of the violent accounts of the wars between Sub-Romans and Anglo Saxons or the Viking and Norman Invasions on the English, you may be forgiven for thinking that the average British or English person had to 'survive' these experiences. Indeed the inevitable complications of warfare; disease, food shortages and lawlessness would have impacted the majority of the population. But the reality appears to have been that a new culture arrives in numbers from the thousands to tens of thousands and rather than eradicating the current inhabitants they establish themselves as a new ruling class or simply settle beside the previous population.

This is often leads to conflict but not in every occasion, becoming the new ruling class would mean conflict with the existing one, and settling would mean competition and possible conflict with previous tenants or nearby communities if on virgin ground. This is particularly true of the Norman Conquest where the relatively small Duchy of Normandy could only provide around 8000 people to control England, a nation with a population around two million. The infamous Battle of Hastings dealt relatively quickly with the English ruling class, with further resistance being limited to London, the North and occasional hold outs that were dealt with within a few years of the initial invasion. It is argued as to whether folk tales like Robin Hood represent a continuation of Anglo-Saxon resistance against Norman methods of rule.

The Viking Invasion on the other hand was more of a long term migration over a period of three centuries interspersed with invasion events. In particular the events surrounding the 'St Brice's Day Massacre' in AD 1002, where the English King ordered the slaughter of Dane's settled within his kingdom (arguably survival was far more the problem for the 'viking' at this moment) and provoked the Danish King Sweyn Forkbeard into a series of lengthy invasions or raids in England until he became King of England in AD 1013. Again the average Englishman suffered loss, shortages and disease due to this, but by the end of the campaigns his royal family had fled to Normandy and he had a new Royal Family to replace them, he possibly had different laws and fashions of dress, new traders to buy from or in the case of Sweyns son, Cnut the Great, a strengthening of the currency and thereby, the economy at large. However, modern British DNA shows no great amount of Scandinavian ethnicity surviving in the North Eastern section of England that was the Danelaw and is mainly clustered in areas that saw long term migration, from the Isle of Man northward through the Scottish Isles but with some clear areas in East Anglia.

To go further back to the establishment of Anglo Saxon populations, again, it is estimated that actual migration only numbered in the tens of thousands moving into a nation that under the Romans had a population around 3 million. Yet again, our historical sources describe the Briton's as caught between death in the form of Saxons or the Sea but instead we find in the archaeological record, people of mixed background buried along side migrants and people of local birth buried with Saxon goods.

So, to answer the question, the majority of people in England today carry a lot of Anglo-Saxon DNA, in remoter areas of the West Country, Wales and Ireland the majority of DNA dates back the 'Celtic' peoples that populated Britain before the Romans and in Northern Scotland, the Scottish Islands, the Isle of Man and some fringe areas of the North Sea coast Viking DNA is prevalent. As to how people survived, it appears that in the majority they tried to get along with the new migrants. Local Britons in Eastern England appear to have rapidly adapted to a Saxon way of life, whereas those in the West took longer and as far as we know, with the foggy (at best) history that survives of the period, fought wars to maintain their Sub-Roman culture. Again this is matched by the conflicts between Danes and Saxons two centuries later, however on this occasion the Saxons were eventually successful in expelling the Danish rulers from their kingdom. Finally the English are today, in an abstract way, under the system and ruling class established by the Norman Conquest. This ruling class held itself aloft for a time by making intermarriage a rarity, speaking Norman-French instead of English and living for the most part in Castles. This system, defining the common people under the 'Englishry' only began to be softened over time by intermarriage but finally crumbled during the Hundred Years War where the Normans lost their french lands and cultural roots, but also found great benefit in trumpeting 'English Identity' against their French rivals.

To jest; we survive today much as we did then, by working, paying taxes, getting along with migrants and avoiding the worst tendencies of our landed gentry.