Here's a slightly different version from The Nursery Rhymes of England (1843) by James Orchard Halliwell:

Bryan O'Lin had no breeches to wear,
So he bought him a sheepskin to make him a pair
With the skinny side out, and the woolly side in.
Oh ! how nice and warm, cried Bryan O' Lin.

Here's another rhyme about Bryan O'Lin from the same book:

Bryan O'Lin had no watch to put on,
So he scooped out a turnip to make himself one
He caught a cricket and put it within,
And called it a ticker did Bryan O'Lin !

Here's a version of the breeches rhyme from The Nursery Rhyme Book, edited by Andrew Lang and illustrated by L. Leslie Brooke (1897):

Brian O'Lin had no breeches to wear;
He bought him a sheepskin to make him a pair,
The woolly side out, and the other side in:
'It's pleasant and cool,' says Brian O'Lin.

There's also a rhyme about Tom O'Lin called Tom a Lin and His Wife and His Wife's Mother.

Thanks and Acknowledgements

The version of this rhyme at the top of the page can be found in The Little Mother Goose (1912), illustrated by Jessie Willcox Smith.