Leeks!!!!

A very nice plot neighbour gave me some leftover leeks last Summer, which I didn't get into the ground until really rather late (end July) as I hadn't a space cleared for them.
I've no idea what kind of leek they are, and not seen my neighbour in a while.
His leeks were harvested in Autumn, mine are still in the ground, about as thick as my finger. They look healthy, no sign of rust or anything... just skinny!! 

If I leave them where they are, is there any hope they'll thicken up a wee bit in the Spring sun so I can at least get them to thumb thickness?

Use them up soon.

Once they start to grow they'll be running to seed as part of their natural cycle. You'll get a solid stem in the centre, which you can cut out, but ir's more messing about and you don't gain anything, other than tougher flesh (the leek, not yours) at the end of the day.