I think I know what the problem is.  When you press the shift key, a 'claw' inside raises a bar which runs across the carriage.  This in turn raises and lowers the platen in a 'cradle' within the carriage.  Someone has probably removed the carriage and refitted it with the 'claw' on top of the bar instead of around it.  This would force the platen up and keep it up.  I think that I can just make out the claw sitting on top of the bar in one of the side view photos.  You will need to remove the carriage and re-fit it correctly.  This is not as scary as you might think.  Pretty straightforward on an Underwood Five.  First unhook the drawband underneath the carriage on the right.  Park the end on the stud provided in the carriage bed.  Remove the two screws at the back that hold the rear way rod on.  At the front of the machine, remove the left hand screw on the scale along the front.  Remove the right hand screw, noting that it goes into a stop piece which will fall out.  Note the position of the piece and remove it carefully as you take the screw out.  Put the screw back into the piece temporarily as a reminder that the right and left screws are different lengths.  Lift the scale up and off.  You can now take the carriage off, comple with the rear way rod.  You should now be able to see the problem.  When refitting the carriage, make sure that the claw is around the rod.  Put the way rod screws in loosely and try the shift to see that the platen does indeed move up and down before going any further.  If it does, reassemble carefully in the reverse order.  Job done (hopefully !)

Thanks thetypewriterman, I followed your instructions and removed the carriage, and replaced it with the 'claw' cradling the bar (some force required). Seems to have solved the problem, the bar just requires some oiling (the carriage at the moment is very stiff to move). Thank you so much for your help!