The Cock, the Cuckoo, and the Blackcock

Once on a time the Cock, the Cuckoo, and the Blackcock bought a cow between them. But when they came to share it, and could not agree which should buy the others out, they settled that he who woke first in the morning should have the cow.
So the Cock woke first.
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"Now the cow's mine!
Now the cow's mine!
Hurrah! hurrah!"

he crew, and as he crew, up awoke the Cuckoo.
"Half cow!
Half cow!"

sang the Cuckoo, and woke up the Blackcock.
"A like share, a like share;
Dear friends, that's only fair!
Saw see! See saw!"

That's what the Blackcock said.
And now, can you tell me which of them ought to have the cow?