OP, what do you like to do? What are your interests? Are you planning on renting or buying when you arrive, or renting and then buying? (I've had clients from the UK do both.) What's your budget? Do you have children, so schools might be important? (Of course, there are a plethora of private schools in Austin to choose from, as well.) Are you more interested in city life or outdoor life, or one of each?
Give us a better picture of you and we can give you a better picture of you in Austin.
Edited due to crossposting: Yes, horse riding is very much possible here - English and Western, driving if you prefer it, most any discipline and seat!
You will NOT miss out on the Texas lifestyle if you live outside of Austin proper. I'd recommend, however, living in one of the small towns outside of Austin to get the "real feel", depending on commute. With luck, you might even be able to have a horse on your place, or live in an equestrian community with trail riding friends.
I'd recommend looking at Georgetown, Liberty Hill (probably the furthest out you'd want to go but definitely a "horsey" community), Leander, some parts of Cedar Park, Elgin (VERY horsey community), Dripping Springs. These are spread out all around the Austin area and each has its special virtues and vices.
You will want to think about the traffic at rush hour if your husband is going to be working "normal" hours (whatever those are these days!).
Welcome to Texas! I loved London and England in general when we visited there briefly (though I confess Scotland was my special love - I suspect it was my genes talking

).