My GF went from essentially a non-swimmer state (only one swimming class, in college, 14 years ago) to OW in 3 months, has finished her AOW, about 40 dives now. She had seen me practicing in the LDS pool and come along to watch several dives, wanted to learn it.
We did this incrementally, once a week visits to the shop pool. Started with wearing a Full wetsuit, plus fins, mask, snorkel, so she was positively buoyant (she has a fear of being a sinker though the staff was pretty sure she wasn't, but needed an edge for comfort), relearning basic swimming strokes doing laps in the shallow end of the pool.
Progressed to swimming laps lengthwise, same gear setup, but she was now going over the deep end.
Then dropped the flippers -- she now had the full suit, absolutely positively buoyant, but had to actually use her arms successfully for the swimming to work.
Removed the wetsuit, had her do laps in the shallow end.
Finally, laps lengthwise, going over the deeper part.
Next, buoyancy -- same kind of evolution, learn to back float in the shallow end, the try it in the deeper part, then learn buoyant on the surface vertical.
The shop staff helped, cheered her on and gave her pointers. I acted as a safety, even in the shallow end during initial things, otherwise I was in water, fully geared, right next to her when she went into the deeper end.
After the above she was fine on the PADI requirements, and her basic water comfort level is quite good now. As others have noted, the underwater part of diving is actually not much classic "swimming" -- the surface part, first normal swimming technique in pool conditions and then adding gear, then learning in worse conditions (cold, chop,...) does take new, building-block practice.
Thats funny, I cannot swim without fins and can't have anything less then the Scuba twin jet fins.
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