Some names I have found that are common on both sides of the pond include: Richard, Scott, John, Jack, Thomas, Alex, Ellie, Charlotte, Sarah, Emily, Olivia, Lucy, and Elizabeth are really just a few. To be fair, there are quite a few overlaps in names on a transatlantic level!
Some differences I have noticed are in the nicknames derived from some of these names. For instance, Charlotte is a fairly common name here and often the nicknames for these girls are either "Charlie" or "Lottie." I don't know anyone by the name of Charlotte in the US who goes by either of those shortened names.
Some names that I have found MUCH more common over here in the UK vs. the US include: Iris, Poppy, Finn, Ollie (or Oliver), Oscar, Mohammed, Ruby, Imogen, Fiona, Harriet, Gaynor, and Harry. On the reverse side of the equation, occasionally my co-workers will comment about a name being "so American!" I always chuckle by these comments but the names where these comments have occurred, include in regards to names such as: Brent, Bruce, Riley, and Ashley. When I think about the names I hear over here in England, I have yet to meet anyone with such a name....
For a general comparison, I pulled the top 20 baby names from 2009 for both girls and boys for the US and UK. If I were more tech savvy, I might be able to arrange the tables side-by-side to one another, but unfortunately I couldn't figure it out! So, you can just skim the lists and compare the lists on your own.
Top UK Boy Names:
Top US Boy Names:
Top 20 US Girl Names:
Top 20 UK Girl Names: