MFA takes 3 working days. So appointment at embassy on a Tuesday, documents in with the MFA before closing time. Might get the rubber stamp end of Friday, if not Monday.
The UK embassy only does appointments on Tuesday and Thursdays. You have to be there in person to swear. Also, the MFA closes at 3pm. This is where an agency is handy, as they won't mess the translation up.
You have to go back to the Courts to get your name changed. You should have done it during your divorce, because ex parte, its more difficult. The process varies by State (as you are US). For California:
If you have a masters degree or above, then the threshold is $45,000. If you don't have a postgraduate degree, then your solution involves getting one.
Beware, there are websites offering IDPs. Small print says they are just a translation. But basically counterfeit. As stated, most rental won't even want to see your licence, but you will sign an agreement saying you have proper documentation, which will implicitly include having a 1968 IDP. Have a wreck, no medical cover, responsible for all 3rd party claims. It could bankrupt you.