If you want to see a rodeo, make your visit coincide with the Houston Livestock and Rodeo. Next year it begins February 26th. It is huge, has first-rate entertainment, and is a good ol' hoot on top of all of that! You should end your visit in Houston so you can go. We never saw so many cowboys in one place.
Here is the website:
http://www.hlsr.com/
You may end up buying a pig to bring back to the UK with you -- some Texas child's school project.
If you want to see wildlife, you should not miss the Aransas National Wildlife Refuge. They have javelinas, alligators, whooping cranes and other unique animals. February is a good month to go because the whooping cranes will be spending the winter there:
http://www.fws.gov/southwest/refuges/texas/aransas/whoopingcranes.html
You can begin and end your visit in Houston by heading to San Antonio on the way out and visiting Austin on your way back in to Houston. You could also do a loop from DFW.
You may also want to include some wineries on your tour of the Hill Country. There are a few near the LBJ ranch.
Note: the bats will not be here (Austin) in February.
Are you planning to visit NY on your way here? If not, I'd look for flights that fly directly to Houston (or Dallas, for that matter) or less crazy hubs. (We've found Cincinnati to be the best hub for transatlantic flights; we've also discovered that there are not that many direct flights from Houston or Dallas.)