I think most names are fairly cross-class.
But there are certainly plenty of names that most people would automatically associate with either the middle or the upper classes. That doesn't mean those names don't ever cross the class divide - just that they're less likely to.
Most of us could write you a list of stereotypically working class names - but nobody will because it would be offensive and unnecessary.
I think some class-less names well used by everyone are:
Amy, Charlotte, Chloe, Daisy, Emma, Ellie, Evie, Grace, Hannah, Holly, Isabel, Jessica, Jasmine, Katie, Lily, Lucy, Matilda, Mia, Olivia, Phoebe, Poppy, Ruby, Rosie, Sophie, Sophia, Tilly, Zoe
Alfie, Archie, Ben, Charlie, Daniel, Ethan, Finn, Harry, Isaac, Joshua, Joseph, Jack, Lucas, Micah, Noah, Oscar, Owen, Oliver, Reuben, Sam, Thomas, William, Zach