a trademark is an exclusive right to trade on a name (i.e. do business using it). Under US law (and I think UK law), certain exclusive trademark rights exist whenever you have an established mark that you're using for business. However, unless you REGISTER the trademark, you can have a hard time making a case in court.
if you're not doing business under the name, then no trademark exists; it's not intrinsic to the use of a name.