Ive built one of Glynn Guest's boats, its a crashboat about 20 inches long, with $50 AUS (20 Pounds maybe) motor and speed control, brushless, 10 cell NiMH, it does 25 Kph which is more than enough for a bit of fun, its sub surface drive, straight shaft and only spins out if I REALLY push it.
What you want to do can be done, If you really want to be cheap buy/obtain a cheap cordless drill, motor, speed control and batteries and will only take a little time to convert the trigger into a quasi mechanical/electronic speed control, no it wont be a rocket but it will be fun.
V bottom boats have been running subsurface drive for a loooooong time with good results before all this surface piercing, flexible drive, latest technology
stuff came out
Nick