Composting of woody material

Yes, I'm sure that it will compost well, as long as you place it in thin layers or mix it up with shredded newspaper/cardboard as you put it in. I'm not a fan of using freshly shredded material as a mulch as I think that slugs and other harmful creepies are attracted to the rotting foliage.

should compost down nicely for next year, mix with brown material as spritz suggests, and turn every couple if weeks or so .

If you can get some fresh horse manure, make a compost heap with pallets with open slats,alternate layers of the manure with a layer of the shredded material. Soak well when filled to the top of the pallets withh water. Leave for two months, turn, when it's' cool, put back into pallets in layers, dressing every four inches with garden lime. Cover the top. After another month, chop out the compost into a wheelbarrow. Use either as a general compost dug into a veg plot, or use as a top dressing on your flowr beds 4 inches deep. Hoddesdon method for general composting always ready within 4 months.
Do not use as a mulch.

I mix shredded woody waste with green material in my compost bins and allow it to stand for a full twelve months - by then everything is rotted nicely. If you have a woodland area, the shredded material is good to use as paths instead of buying in bark chips