Can anybody recommend me one or more UK MBA programs for a low GMAT score? (640 to be exact)

Thank you!

It depends on many factors, the least of which are the other parts of your application (number of years and quality of work experience, undergraduate performance, extracurricular activities, your goals.)

Generically speaking, and not knowing much about the above, Manchester would be a solid choice with an average GMAT of 630ish I believe. It's similar at Cass and Imperial. Warwick would tend to be a better choice for most students but their MBA also recruits students with higher GMAT scores; might be a reach for you.

My GMAT is 640 and I have four years of high quality work experience, including a promotion to team manager. I did relatively well in my undergraduate studies with a GPA of 3.7. I have done volunteer work with my church consistently over the past 3 years or so does that count as extracurricular activities?

Of the schools you mentioned, which do you think would be the best fit? And which would have the best potential for career growth?

Your church work should count towards volunteer work, no doubt.

Warwick probably has the best career outcomes, generically speaking, but without knowing more about your goals, it's still hard to say.

Do you think I have a shot at Cambridge or Oxford?

I think it’s a long shot, as the averages at both of these schools tend to be 50 points+ higher than your score (coupled with the fact that your work experience is lower than average as well, plus that it seems like the other aspects of your profile are not above average.)