When you say "slaves freed in the UK", I assume you are referring to slaves in Britain's Caribbean colonies.
For Britain's Caribbean colonies, it used to be believed that most freed slaves adopted or were given the surname of their owner. However, more recent research shows that, although this did happen, there were other options available to free men and women. Guy Grannum, in his book Tracing Your Caribbean Ancestors lists the following sources for the origin of surnames:
- Surname of an owner - this could be the last owner or a former owner.
- Surname of father - a white master or employee, a freed man, a slave
from another plantation, or the name of the father's former or
- Surname of mother.
- Last forename - many captives had multiple names that were often used
to differentiate between slaves who had similar first names. Many
were surnames of local people and may have been kept as a surname
- Chosen surname - freed men and women could choose their surname,
maybe to confirm family ties, to disassociate themselves from former
owners, or after influential people.
- Given by the church or state for official purposes.
Although men and women could chose their surnames on emancipation, most, if not all, chose surnames from those among the local population. Individuals could (and did) change surnames, and the same individual might be recorded using a different surname on different documents.
Different sources seem to have been more prevalent depending on which island the people came from.
For Barbados, you might find Slave Names and Naming in Barbados, 1650-1830 informative. In this case it is worth noting that
"There was no clear tendency for slaves with surnames to bear the names of their owners or other slave masters".
This has been the source of a couple of "brick walls" in my own research on Barbados!
In the case of Jamaica, the diversity of Jamaica's people and their surnames is reflected in the island's Motto, "Out of Many One People." The inhabitants of Jamaica came from many different countries which complicates the question of surname origins even further.
A useful starting guide here is Jamaican Surname Origins from Jamaican Family Search.
A note about slaves in the United States:
In the United States, there has been some research has suggests that slaves were often known by the name of their original owner. This would not necessarily be their last owner. A useful source here is Herbert George Gutman's The Black Family in Slavery and Freedom, particularly chapter 6. [Note that this book is also available for a 14 day loan on Internet Archive - registration required]
Unfortunately, other sources contradict the results presented by Gutman. In 1783, the British military authorities compiled the "Book of Negroes" from evidence provided by black refugees from the American revolution. The original source material is available in the collection of the UK National Archives (Reference PRO 30/55/100), although unfortunately it isn't available online.
This, and evidence from manumission returns (mentioned in Grannum's book), showed that very few slaves possessed the same surnames as their owners.
It seems that the situation in the United States was just as complicated as it was in the Caribbean.