So in part this depends on who you classify as a 'native' (although the extent to which the 'English' were actually British or not is a topic for another post). I also recently wrote an answer here that looked at the extent and rapidity with which the Normans integrated themselves into the pre-existing English and British elites in the wake of the Conquest.

It's interesting that you used the word 'enslave' specifically, since that's of the major things that does change with the Conquest, albeit not inmediately. Slavery is a major part of pre-Conquest English society, with laws governing slavery appearing as early as the 7th Century. Slaves were a major social caste and appear in legal codes and in wills throughout the period, with a series of laws covering who could and couldn't be a slave, when slaves could and couldn't work, laws on emancipation, and legal protection for them. The majority of slaves - servus - were most likely British, probably from Wales, but were also traded with Hiberno-Norse communities in Ireland, and could also be penal slaves. Interestingly, the early 7th Century laws of Hlothhere of Kent allow for thieves caught in the act to be sold into slavery abroad, while those of rough contemporary Ine of Wessex forbid the selling of English slaves overseas. The one thing we don't have is any indication of precisely how big this social caste was. At the time of Domesday Book in 1086, some two decades post-Conquest, slaves still comprised some 10% of all listed households. The institution appears to have gone into decline after the Conquest, and by the time William of Malmesbury writes retroactively about the plight of slaves in Bristol in the 1120s, slavery seems to have been all but stamped out entirely. Prior to 1066, therefore, it's generally held that slaves were an even greater proportion of the population.

There is a general decline in the number of Freemen in the wake of the Conquest, although Freemen shouldn't be conflated with free men. (All Freemen are free, but only some free men are Freemen. Confused? Good. Welcome to Medieval Studies, let's not even get started on the mancus.) Freemen were already less common outside of former Danelaw areas (a phenomenon itself the subject of some debate as to what precisely that represents) and do appear to get scarcer by the twefth century. At the time of Domesday, This doesn't necessarily represent an enslavement as much as it does a demotion in status and the imposition of a landlord. While this would have been far from popular or, indeed, just, it's far from enslavement. Despite the popular modern misconception that "serfs" were property or essentially slaves, a medieval tenant farmer was no more a slave than a modern tenant, or employee of a large corporation (Of course, this point is itself up for debate to those with a more Left Wing political outlook, but that's largely academic to this particular point).

For the everyday English tenant farmer, whether villager or smallholder (some 34% and 30% of households at Domesday respectively), the Conquest of 1066 is unlikely to have had that significant an effect on their daily lives, with a few important caveats. William was forced to spend around two decades fully pacifying England, particularly in the North. In 1068, facing an embarrassing defeat by the people of Exeter, William negotiates and reconfirms the pre-Conquest rights and privileges of the city and its burghers, for example. Where things did noticeably change on a day-to-day basis was in law and order, and the forests. English law was orientated largely to keep the peace through the suppression of feud and restorative justice, at least theoretically. Æthelstan repeatedly complained of his lords having thieves hanged, for example, when he laid out clear punishments in his legal codes. Norman law, on the other hand, was far more punitive, and often replaced fines - such as those stated by Æthelstan for theft - with punishments such as mutilation. A convicted thief, for example, might lose a hand, a punishment previously restricted to moneyers who defrauded the coinage.

The other important change was the very unpopular Forest Laws, which greatly curtailed the ability of the common people to forage and, more crucially, to hunt in forests that were now made part of royal or manorial demesnes. Excavations of sites like the butchery at the Stafford burh suggest that game made up a significant addition to the pre-Conquest fresh meat supply.