It doesn't need to be anywhere near as complicated as all that.
To start using your new name officially in the UK:
- Just start using your new name. Contact your bank/utility companies/store cards/anything else with your old name on and ask to change it. That's all you need to do.
You do NOT need to change your name on your passport unless you are desperate to, as it can be a hassle to do it. There is no issue at all with keeping your passport and visa in your maiden name, but using your married name officially for everything else. You just need to make sure you book any travel in the name on your passport, and if you need to show ID for other things with your married name on, just show your passport plus marriage certificate.
If you do wish to change your name in your passport, it would have been best to do it BEFORE you applied for your FLR(M) visa, because if you change it now, you will be required to change in on the BRP within 3 months (or you can be fined £1,000)... and it will cost you £107 and you will be without your passport and visa for a few weeks while it is processing.
So, if you would prefer not to do that, just wait until just before your next visa application to change it in your passport, so your visa is automatically issued in the new name and you don't have to worry about changing your BRP.
Not sure what you need to do about changing your name in the US though (i.e. on your SS card etc.).