What is the admissions process for top UK graduate programs? What is a considered competitve resume/application? I'm considering applying to a few top UK universities for a PhD in Engineering, for the 2016 year.  I would really like to attend a strong program, namely, Imperial, Cambridge, or Oxford.  


Admissions to undergraduate courses at these universities are extraordinarily competitive.  How does the graduate admissions process compare?  Is it just as competitive, or is it more relaxed?
What credentials would an American applicant need to have (grades, research experience, job experience, etc) to be competitive?
Do you need to have both an American BS and MS degree to apply to a UK PhD program?  Or is merely a BS sufficient?
UK PhD programs pretty much universally require a MSc or equivalent so I would expect a MS to be required. For some lower ranking universities you may be able to make a case based on the differences between US and UK bachelors, but I think even this is doubtful.

Thinking about it this is not entirely true. There are some doctoral training centers which offer combined masters and PhD courses for which you don't require a Masters, but outside these a masters is required.

Academically, I expect top UK institutions have similar requirements to top US institutions. Specially, they will require a 2:1 or equivalent, with some top universities requiring a 1st. There is no simple conversion to GPA or other systems. But, roughly speaking at 2:1/1st corresponds to 3.2/4.0 in GPA.

However, academics are mainly just a minimum requirement. The main determining factors are the enthusiasm you can show for research, particularly related to the area you are applying for. Most UK applicants will have at least some research experience from experience from their course and the majority will have also done a summer research job or similar. 

I would still expect applications for to places to be quite competitive. Less in terms of raw numbers compared to undergraduate, but more the quality of applicants. Some interesting numbers for Cambridge can be found here. About 1/3 of applicants get an offer and about 1/4 applicants start the PhD (the difference presumably fail to make their grades or take another offer). This is much higher than undergrad, where a low estimate would be 1/10 get an offer.