Can I swap a hard drive from a Dell to a Compaq without needing any other bits and bobs?
As the hard drive has died in the compaq and the screen has died from the Dell.

Perhaps. The issue is whether the Compaq PC has the correct form factor to accept the Dell's hard drive. If the PCs are about the same age you may get lucky, but if one is older than the other probably not. I suggest you open up both machines ansd see if the form factors are the same.