You can use your new name in the UK without changing it on your passport - you just start using it.
However, your visa can ONLY be issued in the name on your passport, which means you can also only travel in the name on your passport/visa.
The name you use for the visa is irrelevant and many people end up just keeping their maiden name on their passport and all their visas.
It is a massive hassle (and expensive) to change your name on your passport and visa once you are in the UK (unless you wait until just before a new visa application). And, if you wait until citizenship, you have to get your citizenship in your maiden name, apply for a UK passport in your maiden name, then change your name on your US passport, and then change your name on your UK passport... which again is a hassle.
So, if there's any chance you might wish to change your name on your passport in the next few years, I would advise changing it in the US BEFORE applying for your spousal visa.
If you do that, your visa will be issued in your new name and you won't have to worry about changing it in the future.
It only takes a few weeks for a passport name change in the US (or I believe you can expedite it), so I would take the time to do it before applying for the visa, so save you time, hassle and expense later on.