A couple of years ago I started walking at our local track every morning with the retirees. I felt very sad that they were all passing me when I was half their age. But I could barely manage 3 miles in an hour, and running for a full minute wasn't even a possibility. My second 5K, I was passed by a older man with one leg, on crutches (seriously!
).
I finally started (and eventually finished) the C25K program last October. Here's the proof of the progress I made: Today I finished 5 miles (4.25 running/.75 for warm-up cooldown) in 58 minutes. I can remember about Week 4 of the C25K when I hit 4.3 on the treadmill, I felt so fast.
Considering I started walking at about 3.0, I guess I was, at least for me. Last time I was on the treadmill, I managed to do a few minutes at 7 mph, and was comfortable at 6mph for quite a few minutes at a time.
I guess my point is that it's going to take time, but you will get faster, even without trying. Don't push too hard, or too soon though. If you strive to be consistent (by that I mean don't abandon running because someone else is faster), you will make progress.