How would you bump start a car without turning the key to Ignition? Or do you mean holding the key to crank the starter at the same time as letting out the clutch?
Jamie, get a £5 multimeter and check the battery. (before charging it)
12.4v or below and its either battery or alternator.
Jump or bump it, if the voltage does not rise to 14.4v whilst running, the alternator is dud/aux belt slipping/broken earth strap are the likely culprits.
If 12.6 or 12.7v, the battery and alternator are healthy and it could be a starter issue, try putting it in gear and rocking car back and forth to move the drivetrain slightly and try to start again.