Is it possible to find out how long your lord / hero will be injured?

There are a variety of ways your hero or lord can be injured.
From injuries in battle, to being the target of an assassination or having failed your own assault on the enemy beasts.

However, once injured he's out of commission for a while.
He's getting some well deserved rest, while the rest of the world continues its perpetual war.

How and where can I find out when the coward in question will be ready to leave his bed and return to glorious combat?

Yes you can, when a lord is assassinated. The game will force you to pick another lord, you can then in the lord selection scroll down and see the one that is killed if you hover over it you will see how many turns remaining.

You can do the same when recruiting a new army, and for hero's while recruiting a new hero you can see the wounded ones on the bottom. Hovering over them will also tell you how many turns remaining.

If you open the new army interface and hover over your lord. You will see XX turn remaining before you can select Lord.

XX being number of turns before Lord become available again.