...well, what if you don't change anything in the US, but want to use your married name in the UK? That can work as well, right?
This is what I have done. I married in the UK, and couldn't be arsed to change things back in the US. My US passport and UK visa stamps in the passport are all in my maiden name, but all my other UK stuff - bank accounts, taxes, etc., are in my married name.
In the UK, you are entitled to use either your married or maiden name as you wish.* Your marriage certificate is proof that you are who you say you are.
Works fine for me.
I have never had a problem with showing my passport (with maiden name) + marriage certificate to identify myself under my married name.
*As long as you are honest and don't try to be deceptive. For example, the IRS and US Treasury know me under my maiden name. I have UK bank accounts in my married name, and I report them to the US Treasury every year as required (I suppose I could possibly get away with hiding them from the US by claiming that I was a different person, but that would be wrong and illegal.)
When I had my old passport everything was in my maiden name I had to provide my new passport to have my name changed to my married name. .
I don't know what you mean by having your name changed. If I wanted someone (bank, credit card company, utility company, NHS, HMRC etc.) to start calling me by my married name I just sent them my marriage certificate and they changed my name on their computers. Easy peasy.