How do I handle academic references for US PhD program, when I have been out of academia for a (very) long time? I live in the US and have been working as an engineer in industry for over 10 years. I graduated from a top UK University, with outstanding grades in a Masters degree and several university-level prizes.

I have decided that I want to pursue a PhD, as I have a strong desire to be doing more advanced and challenging work. So, I am planning to apply to the PhD program of the best university that is in my area (which happens to be a top-tier US institution). My family and I are fairly settled in the area and we have no desire to relocate.

The institution has an admissions requirement that I have to provide three recommendation letters, two of which should be academic. Therefore, it seems that I have little option but to reach out to some of the professors that I worked and studied under during my undergrad course 10+ years ago. I am planning to provide one industrial reference and two academic; however, I am concerned that my undergraduate course was so long ago that the professors will struggle to remember me, given the hundreds of undergrad students they interact with each year. So, at best, even if they do generously agree to write a letter, it will probably be fairly generic and unspecific. And of course, it won't be in any way a good reflection of the more experienced and mature person I am now.

At worst, they might simply refuse, given the amount of time that has passed. I wouldn't blame them, and in a way I feel bad asking, after all this time.

So, what should I do in this situation, given the amount of time that has passed? Is there another way I can approach this?

The irony is that, given my industrial experience and personal development over the past 10 years, I think I am almost certainly a stronger candidate for a PhD program now than I would have been straight after I finished my undergraduate degree; however, given the amount of time I have been out of academia, obtaining strong recommendation letters is going to be much more challenging.
In contacting your old professors you might start by reminding them of what courses you took with them and what grades you earned in those courses. It would also be appropriate to supply a copy of your full academic transcript.  

I frequently get requests like this from students that I haven't seen in 10 years or more.  I comment on the students transcript and grades in my courses and whatever I can remember about the student.  This isn't ideal, but it's better than nothing.  

Another common strategy for students who've been working in industry for a long time is to begin graduate study by taking a few courses on a part-time basis at a local university.  It's much less of a commitment by the department to allow you to take a course or two than it would be to admit you as a full-time student and provide you with financial support.  If you do well in those courses then you can ask the instructors for academic references.