I haven't trained divers, but I have taught an awful lot of people to swim (kids and adults). One of the most prevalent problems was always how to overcome "bicycling". The easiest method is to put the swimmer in a pool, have him put both hands on the side, get him/her into a horizontal facedown position (a snorkel will help with this...you don't want the head and shoulders high), and have them flutterkick, as if they were trying to push through the wall. Now the hard part (for you). Stand next to them and place your extended arm under their KNEES, about a foot or a little more below the surface. They are not to touch your arm. Bicycling takes place high in the leg, at the hip, with their knees bent. If you do not allow the knees to bend and keep the swimmer stretched out, the knees will be straighter and they go into a more natural up-and-down kick. Of course, you also have to work at keeping their knees and ankles supple and not locked, but that comes with practice.
This also works when you have them flip over and do the same thing on their backs. The difference is to put your arms under their lower back in order to keep them horizontal. Now, the instruction is to keep the knees more or less in the water (it's hard to bicycle because it's more difficult to lift their knees clear of the water). Having them point their toes slightly (again working on keeping the ankles supple and not locked) and try to kick water upwards with their fins, like kicking a soccor ball. When they can do these motions stationary reliably, put them on a float or kickboard and have them motor around the surface of the pool for a while. Then do the same thing snorkeling a few feet below the surface. Progress on to doing the same thing in full gear.
It takes time and practice for this to become natural. However, I've had great success with it in remarkably short periods of time. Once this motion becomes natural, there should be much less conflict with the coordination of hands and arms (though ideally those should have a much reduced role in diving motion).
Hope this helps.