Both Normans and Bretons were granted lands and position following the Norman Conquest. There were already a number of Normans in England before the Norman Conquest, the court of Edward the Confessor contained several, Edward himself had spent some time in Normandy, the first Norman-style building in England Westminster Abbey was built before the invasion.


Very few, apart from some lords & their families. With a huge majority becoming ‘English’ by virtue of the fact that integration was the only practical option in running their lands & after a generation or 2 was probably inevitable – England became their land & the English their countrymen. The Norman were not French but Norman or ‘Men of the North – Norsemen’ who had a less then friendly relationship with the French. King William himself died in battle against the French. French settlement in England came from being given asylum by the protestant English to French protestants being victimised in Catholic France. This is also true of other European protestants who settled in Britain & British North America.
It all depends what you mean by ‘a lot’. There were certainly a large number of Norman and French noblemen who were granted estates in England and settled. Indeed the English (or rather Anglo-Danish) aristocracy was almost entirely displaced. However, there was not a significant movement of people of lower classes who of course made up the bulk of the population.
The main migration of French people to England in fact happened in the 16th century when some 50,000 French Protestants, known as Hugenots, fled to England to escape religious persecution.
Not that many, it was mainly nobles and a few servants, perhaps a couple of thousand.
And vice versa. Not many.