Although they may have considered doing so, as far as I can make out, Mary Seacole and Thomas Day never actually established their business in Aldershot.
I assume that many people will already be familiar with Mary Seacole's memoir, Wonderful Adventures of Mrs. Seacole in Many Lands, and so will be aware that the sudden end of the Crimean War and the evacuation of the British Army was a financial disaster for Mary Seacole, and she was left virtually bankrupt.
From her memoirs, it seems that she was one of the last to leave the Crimea, departing sometime in July 1856, but she
"... did not return to England by the most direct route, but took the opportunity of seeing more of men and manners in yet other lands."
We know that Mary Seacole arrived back in the UK in August 1856. Again, from her memoirs she says that:
"... we set to work bravely at Aldershott [sic] to retrieve our fallen fortunes, and stem off the ruin originated in the Crimea, but all in vain; and at last defeated by fortune, but not I think disgraced, we were obliged to capitulate on very honourable conditions. In plain truth, the old Crimean firm of Seacole and Day was dissolved finally, and its partners had to recommence the world anew."
However, the bankruptcy notice published in the London Gazette on 28 October 1856 makes no mention of a business, or even an address in Aldershot:
The current addresses for both Mary Seacole and Thomas Day are in London. Their prior addresses were in the Crimea. So I think we have to conclude that they probably attempted to establish a business in Aldershot (as suggested my Mary Seacole in her memoirs), but that they were unable to do so before their creditors filed for bankruptcy against them.
Copies of the bankruptcy notice were also subsequently filed in
The Edinburgh Gazette on 31 October 1856
The London Gazette on 9 January 1857
and The London Gazette on 6 February 1857
The address details for Mary Seacole and Thomas Day are the same in all the notices.