Hi and welcome! I presume this typewriter is new to you? It sounds as though it's in generally not great condition (your pictures won't open, no doubt because you need three postings in order to be able to attach pictures).

I am not really familiar at all with the old Underwoods, but there are some general things you can check. I wonder whether the ribbon advance mechanism is just in need of cleaning -- on some machines, the ribbon mechanism is coupled to the carriage's movement, but I don't think that is the case with the Underwoods. As for the carriage not advancing, the first thing I would check is whether the machine's mainspring has become detensioned, or simply is broken. Or perhaps the strap that connects the spring to the carriage (and pulls the carriage from right to left) has become detached, or is missing entirely.

Others here with more knowledge will jump in, I am sure.

Thanks - hopefully three replies to thread will allow me to post photos. Pretty crucial

Here's a link to my photos
https://drive.google.com/open?id=0BwmrHl56AGVZdHRTMHRaMFFFQUU

If you pull the carriage gently to the left and type with the other hand does it advance correctly?  If it does then it is some problem with the mainspring and strap - maybe in place but not tensioned - and if it does not then there is some issue with the complex chain of events involving the escapement which releases the carriage to move over one space with each keystroke. You hope it's the former.

I would not conclude the ribbon advance was not working until I got the carriage movement working - ribbon advance depends on carriage movement.  As for "the rod" typewriters are full of them so I might have said "a" rod vs. "the" rod.   Not sure what the one you picture is connected to but I notice a number of collars with set screws so it's possible one is out of the position which otherwise would have limited the rod's side to side movement.

I was able to open your photos externally and good photos, and yes, the typewriter is very clean and good looking!
 

Thanks Repartee - I have retensioned the main spring and checked the position of the strap and I can feel the carriage is under tension now as I move the carriage back and forth. I dont think I realised how much more tension I could get from the spring so hopefully thats helped things. However, even though the spring tension is better when I push the carriage to the left as I type with the other hand I'm still getting no movement. So I fear it is something inside linking the escapement to the carriage. 

Try as I might I can't see how its supposed to work and suspect I have something missing. When I pressed the space bar I can the rods connected to it move but do not come in contact with anything. I have taken a photo of the rods I'm referring to (marked in red).

https://drive.google.com/open?id=0BwmrHl56AGVZanRidjVXOWgzX1U

Also, there is still the issue of the 'loose rod'. I have taken some more photos indicating where its positioned. I've also removed it so you can see what its connected to. What does this do and why does it come loose? The collars do not seem enough to hold it in place.

https://drive.google.com/open?id=0BwmrHl56AGVZUnZ4anM0dHU4VHc

Sorry its a lots here! But thought I'd include it all in case you can spot anything wrong :-)
Nicola


 

From what I see, it looks like either you don't have the ribbon mechanism advancing the ribbon without that part, or else you have no way of releasing the shift lock without it.  As for the space bar lever in the back, it may need to be adjusted in order to contact that part.  I have repaired lots of Underwoods, and I will check one of mine out to see just what these parts exactly do, and, if my suspicions are correct, that loose one has an adjustment to make it fit more tightly within its area.  I'll report back tomorrow.