Absolute rigging fails as my mesh is distorted Really new to Blender and 3D modelling as a whole but as it's part of my college module I really need to learn it quickly.

Anyway, I was following a tutorial on rigging and I believe I followed the instructions precisely, however when I apply the armatures to the mesh and move them around in pose mode I end up getting some sort of Junji Ito styled abomination. Some help would be really appreciated!!

This is how the model should look with the bones overlaid:





And this is an affront to humanity.
Well from the second image your issue is that not all meshes are parented to the bones.

First off, use the mesh with already applied weight painting. Go to weight painting option when selection the mesh and then see where the weights of bones are applied under Vertex groups (downward triangle).

Go through each bone and if you added automatic weights, edit them by selecting Draw or delete extras with Subtract brush on top left.