The "1848" OS map is a little misleading. At the very bottom it says "New Railways, Houses, &o, inserted in 1864". And since the Museum was only handed over the keys on 18th October, 1860. It would seem that this building was added in later.
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Checking Gores [1821 & 1843] I found two entry which refer to "Mill Place, Shaw's Brow". So it was either on the site of the museum/library, or the vegetable market? The 1848 OS was engraved in 1850. I would check with LRO, as sometimes they have the earlier editions. If they have 1850, I'm sure you'll find 'Mill Place' located on it.
Good luck.
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