I have posted this brief history of surnames in Europe before.

The issue with answering your question directly is that the transition happened locality by locality over many centuries, so every place that it happened, there would be a bit of a different reaction.

New York history is my area of expertise so I can answer this at least for there. When the Dutch settled New York in the 1620s (or, "New Netherland" as the colony was originally called) the Dutch still used patronymics. So all the Dutch, Scandinavians, and some of the Germans living there had two last names. If your name was John and your dad's name was Peter, then you'd be John Peterson, or "Jan Pietersen" in Dutch.

But you'd also have a second last name because there would be other Jan Pietersens living nearby. So your second last name would be a descriptor--"John Peterson of the Hill" for example. Or, in Dutch, "Jan Pietersen van der Bilt". Another John Peterson may have come from the town of Buren back in the Netherlands, so the John Peterson "of Buren" was known in Dutch as "Jan Pietersen van Buren".

The English conquered the colony in 1664, the Dutch got it back in 1673, and then the English regained it for a hundred years beginning in 1674.

Around 1680, the English government. began to force the Dutch population living in the New York colony to drop the "Dutch name" patronymic custom and start using English style fixed surnames.

A few families got to keep their patronymic last name, such as "Pietersen", "Andersen", etc., but very few because the English policy was, if multiple families could claim the last name, then all those families had to use something different.

Much more common was to use their second last name (which is called a "cognomen"). This is where a name like "Vanderbilt" came from ("of the Hill"), or "Dewitt" ("the White") or "Van Patten" (Patten or Petten being the name of a town back in the Netherlands). Some are less obvious, like Humphrey Bogart's last name. His ancestors lived in Brooklyn at that time and owned an orchard, and the Dutch word for orchard was "Bogart". Same with the Roosevelts-- "Van Roosevelt" means "of Rose field".

Yet, other families shared a cognomen with another family as well and had to come up with something else entirely. And some simply refused and had a name assigned to them by the English government against their will.

There are many instances between 1680-1710 where a person began to be recorded with their new last name in English/British government documents while still continuing to use their old last name in Dutch church records.

Once this generation of New Yorkers' kids grew up, who were bilingual and more familiar with English culture, they accepted the new last names without much fuss, with many families anglicizing their last names a bit in the early 1700s.

So yes it was a bit of an awkward transition at least there in New York, because many Dutch people didn't want to do it, and many changed their names entirely so that they were known as one thing to their church and neighbors, and known as something else to the government.

In other places back in Europe when similar transitions happened, there are often historical records indicating that the put-upon commoners often resisted the change. I don't know of any place where commoners ever elected to adopt last names themselves. As far as I have ever read, it was always mandated by the lord/government, presumably because it made taxes and generational debts easier to keep track of.

In New York at least, this led to another issue because the Dutch had a strict first name naming custom as well, which initially went unaffected by the English mandate but crept up over the next couple generations. This led to the adoption of "middle names" by the mid-1700s by Dutch New Yorkers, a custom that became a widespread practice throughout the U.S. by 1900, whereas in Britain, people continued to have only a first and last name until fairly recently. More on why all that happened, you can read in another post I wrote here.