Model A+ for size and power consumption.
USB memory for video storage for easy transfer.
Waterproof housing for an action camera, from eBay, for cheap waterproofing. This had the added benefit of waterproof button actuators built-in that you may be able to use to control recording.
The relative merits of the scouts making the thing vs. using it are for you to decide, but be aware that HD helmet cams with accessories, battery, waterproof housing etc are now available for Pi money (£30).
Bear in mind that the Pi/ PiCamera lacks audio input so you will have silent videos if you go that way.
An option would be to use a cheap USB webcam with a Pi Zero. The cable is better suited and you get audio.