The much-quoted extract from Moore's Rental of Liverpool 1667-8 ... "Sugar-House Close ... This croft fronts the street for some twenty-seven yards and I call it the Sugar House Close, because one Mr Smith, a great sugar-baker at London, a man as report says, worth forty thousand pounds, came from London to treat with me. According to agreement he is to build all the front twenty-seven yards a stately house of good hewn stone ... and there on the back side, to erect a house for boiling and drying sugar, otherwise called a sugar-baker's house ..." (VCH)

... an unreferenced sentence from Hugill's 'Sugar and All That' ... "Allan Smith, a 'great sugar baker' of London, transferred his activities to Liverpool in 1666 after the Great Fire, which started in the area of his bakery - though not caused by him."

... and Professor Ramsey Muir in his 'History of Liverpool' states that it was built about 1668, by "a Mr Smith, a great sugar-baker of London. He rented from Sir Edward Moore a piece of land in Cheapside on the North side of Dale Street, on which he erected a building forty feet square and four storeys high."

... however, there appears to be no evidence that this sugarhouse was ever built ... and Mr W F Irvine was much more guarded in his conclusions. He says, "It is not clear whether Mr Smith ever completed his bargain," and in his view those who read the 'Rental' will concur. (from Transactions of the Historic Society of Lancashire and Cheshire 1907)

... yet the same journal states that "John Danvers (brother of Daniel Danvers of Liverpool) married Susan, daughter of Allyn Smith, of Nether Pillerton, co. Warwick, and was in partnership with his brother-in-law, Allyn Smith the younger, a large sugar-refiner, of Battersea. It was this Allyn Smith the younger, who bargained with Moore ... in Liverpool."

Now, if the Smith children were baptised as infants, as was usual and often essential at that time, then I think this last sentence improbable, for Allyn Smith the younger would only have been 19 years old in 1668 and far too young to be a great sugar baker and to be bargaining with Moore.
Whilst I have not yet found the necessary documentary evidence, I consider that the said 'Mr Smith' was one of three ...
* Allyn Smith's father, Allyn Smith (1607-1673), though whilst he appears a wealthy man I've nothing to show he was a refiner;
* much more likely Allyn Smith's uncle John Smith (1621-1684) who certainly was a refiner in Battersea, and also the only verifiable evidence, the Rental, does not give a forename;
* an outsider, an Allyn Smith, b.1701/2 to a John Smith of Battersea, mentioned by Macnamara in his Memoirs of the Danvers Family, however as yet I've nothing relating this family to sugar or to the Warwickshire family.
I don't think any of them went through with the bargain with Moore, but it may be that within a few years one of them set up in partnership with Daniel Danvers and Richard Cleiveland in building a refinery in Redcross Street, or assisted Allyn Smith the younger in doing so (though I haven't found any verifiable reference to him being in Liverpool). Geoffrey Fairrie says in his Sugar Families book, "The first refinery in Liverpool was built in 1673 by Smith, Cleiveland and Danvers. The Smith family were sugar bakers in London, but business was dislocated by the Great Fire of London in 1666, so one of the sons, Allyn, came to Liverpool and started a refinery with his two partners on the corner of Redcross Street and Preesons Row."

Whether or not Allyn Smith (1607-1673) was refining in the City close to the starting point of the Great Fire of London in 1666 I've yet to establish. The family seat was Pillerton Hersey in Warwickshire. His will, written 1673, refers to his mansion there but to no other premises, and son Allyn (b.1649) was given just £10 implying that money/estate had already been settled upon him. We can say with certainty that his brother John Smith (1621-1684) refined in Battersea in 1675 and probably much earlier. In his will, written 1682, he left his two sugarhouses and all other premises in Battersea to his wife Anne, and after her death to his 'cousin' [nephew] Allyn Smith (1649-1723) the son of his aforesaid brother.
At this time the young Allyn Smith was refining in Bristol, where in 1674 he married Ann, a daughter of the much respected refiner Anthony Wood (two other daughters married refiners Morgan Smith of Bristol and Samuel Buttall of Plymouth/Topsham), and they had four children. In 1680, following the death of John Danvers of Battersea, who had married Allyn Smith's sister, Susanna, there was an indenture/settlement between John Smith of Battersea, Allyn Smith of Bristol, John Neale (Allyn's brother-in-law) and four others including Robert Bristow a London grocer whom the widow Susanna Danvers was to marry. After the death of his first wife, Allyn Smith married widow Mary Hook, nee Wear(e), in Bristol in 1682, and they had three children. Allyn Smith had moved to Battersea by 1693, where he and Samuel Danvers took out a mortgage for £2000, and in 1696 he and Danvers, along with Susanna Bristow and her son Daniel Danvers, entered into an agreement regarding properties in Norfolk and Suffolk.
It's probable the connections between the Smith and Danvers families were at first social as Anthony Danvers, along with his father, purchased the manor of Horley in Oxfordshire, just 8 miles across the fields from Pillerton Hersey. Of his children ... John Danvers (1624-1678) married Susanna Smith and became a refiner in Battersea, Samuel Danvers (1636-1709) also refined in Battersea, Daniel Danvers (1644-1710) became a refiner in Liverpool, and Susanna Danvers (1638-1685) married Richard Cleiveland merchant and refiner of Liverpool. It would appear that the family eventually left Horley for Holborn, London, around 1653. Both John and Daniel were educated at Winchester. Regarding John Danvers, Macnamara further suggests, "Eventually he became a sugar-refiner at Battersea, and it may be that was partly by his marriage that his career was settled. Anyhow, we know that he married Susan, the daughter of Allan Smith, of Nether Pillerton ... and that he was associated at Battersea with Mr Smith, the great sugar-refiner, who ... introduced the sugar-refining business to the town of Liverpool." ... his close neighbour was indeed John Smith, who, in 1675/6, along with John Danvers and Godfrey von Itterne of Bristol purchased the lease for property in Sulgrave, Northamptonshire, again very close to Pillerton.
John Danvers died in 1678 having recently purchased house, farm and lands in Essex. Most of his estate went to his wife and his only surviving son Daniel, however having bought the copyhold sugarhouse along with his brother Samuel, the agreement stated that Samuel was expected to purchase his brother's half share from the estate, thus keeping it in the family. Daniel worked at the sugarhouse, but it's history is not clear as Samuel appears to have been in partnership with a John Smith by 1709. In that year they were examined for bankruptcy, as well as it being the year Samuel Danvers died.
Allyn Smith the younger died in 1723. He left his sugarhouses and premises in Battersea to his sons Allyn and Joshua and appointed them his executors. Both married in 1724, Allyn to Mehetabel Hasell and Joshua to Ann Ludlow, though neither marriage appears to have produced children. When Allyn died in 1738 he left small legacies for his bookkeepers, mourning for his boilers, and rings for each of his sugarhouse servants. Joshua continued running the business and is listed in the trade directories as late as 1766. He died in 1770 having written his will in 1765 mentioning freehold and copyhold estates in both Battersea and Sussex, leasehold estates in Battersea, and manors in both Warwickshire and Gloucestershire, but made no specific mention of the sugarhouse(s).
The Collage image 6720 shows the extent of Joshua's property in Battersea around 1750.

Reading ...
1. Transactions of the Historic Society of Lancashire and Cheshire 1907, pub. 1908.
2. Memorials of the Danvers family (of Dauntsey and Culworth) by F N Macnamara, pub. 1895.
3. Wills, summaries of which can be found on my Wills page.