The US Consulate visa application process is pretty automated. The first thing you must do is do the US Visa application online and print it out. A unique barcode will be printed on your application.
Electronic Visa Application Forms - Instructions Page
Then, after (and only after) you have gotten your US Visa application completed, you can go to the NIV Appointment System.
NIV Appointment System
Simply follow the instructions. It will tell you what dates and times are available. Given your situation, book for the earliest one available but if nothing is available soon, don't dispare necessarily. Check back from time to time (appointments sometimes open up). There are options to change your appointment, just make sure to cancel your old one if you reschedule!
Typically, the US Consulate wants you to apply for a visa in the country which you are
residing, in your case, Saudi Arabia. I doubt that trying to apply in the UK would be quicker or easier for you, and worst case, your application could be refused.