There IS a carriage lock.  It is a little chrome plated lever under the carriage, but I can't quite remember if it is on the right or the left !  Failing that, look underneath the machine to see if the draw cord is tangled up around the spring drum.  This is a very common fault with these machines, caused by the grease inside the spring drum hardening and causing sluggish action - leading to the cord becoming tangled up.

I thought so !  That metal spring (like a clock spring) was packed in grease when it left the factory.  The grease has now set like concrete.  The spring drum (that you call a wheel) needs to be dismantled and the spring taken out.  Clean spring and drum with solvent (carburettor cleaner should work),lubricate with SEWING MACHINE OIL, reassemble, reattach the draw cord and adjust for correct tension.

Now I'm wondering if the spring is in fact the problem. It appears to have a lot of oil on it and doesn't seem to be stuck or congealed. There's a hook underneath the spring drum that I assume is designed to prevent the drum from moving back when it's under spring pressure. This hook seems to be preventing the drum from turning at all.
     Here's a picture of the drum:



And here's a picture of the same place with the drum removed, showing the hook:



How is this hook intended to interact with the drum, and is there a way that the hook is causing this problem? There's a gear on the bottom of the drum (meaning on the bottom of the drum when the typewriter is upside down; the gear is between the drum and the typewriter); this gear meshes with the hook.

The hook allows you to set the spring drum tension after you have refitted it.  See that little stud next to it with the screwdriver slot ?  You need to remove that to allow the pawl to work.  You turn the spring drum with your fingers using the little raised parts to grip it, and as you do, the ratchet operates undeneath.  If the tension is too great, you can let a little off by flicking the pawl.  When you are hapy with the setting, you replace the stud to prevent the pawl moving when the machine is in use.  I am 95% convinced that the problem is congealed grease.  This problem is unique to Imperials.

I've removed and reinstalled mainsprings without much problem and didn't really do anything special. Just hook the end of the spring in the drum and then slowly start coiling the spring while using a thumb to keep it in place. Of course spings might vary from machine to machine, and I don't think you mentioned which typewriter (make/model) you're working on, so it's difficult to be specific.