The expression "I'm a bit crook", meaning "I'm not feeling well", is quite common usage in Australasia. Can anyone enlighten me as to its provenance?

Any possible relation to German "krank"?

From a Gaelic dictionary:
crog
an aged ewe; from the Scottish crock; cf. Norw. krake, a sickly beast, Fries. krakke, broken-down horse, etc.

I found a slang site that says: "Crook : sick, or badly made". Maybe it started with the latter and spread to include the former?

re:"Crook : sick, or badly made".

as in
there was a crooked man
who walked a crooked mile
down a crooked path
across a crooked stile
took a crooked wife
built a crooked house
and forever after,
lived a crooked life?

Crooked means grumpy, bad-tempered, in Newfoundland English. I suppose being sick (ill) could make one crooked but the Dictionary of Newfoundland English doesn't give a derivation for this word, so no clues there.

I'll bet it's related to the bent thing: when you're sick, you're often bent over, either vertically or horizontally.

there was a crooked man
who walked a crooked mile
down a crooked path
across a crooked stile
took a crooked wife
built a crooked house
and forever after,
lived a crooked life?

Just for completeness' sake -- the version I grew up with went

"There was a crooked man
Who walked a crooked mile.
He found a crooked sixpence
Upon a crooked stile.
He had a crooked cat
Which caught a crooked mouse,
And they all lived together
In a little crooked house."

There was a crooked man
Who walked a crooked mile.
He found a crooked sixpence
Upon a crooked stile.
He had a crooked cat
Which caught a crooked mouse,
And they all lived together
In a little crooked house.

This is closer to the version I remember, but I seem to recall something about a crooked (walking) stick.

Me too, doc!
The crook (ailing) gives the poem a whole new meaning ... and makes more sense in an odd sort of way.