I work in publishing, though much smaller print runs than we're talking about here.
Eye Books will have made the decision how many books to print initially, and how many to (immediately!) reprint. There's no info in the book about which printing company they've used.
I think, from looking at Amazon, that Amazon orders for this title are also fulfilled by Eye Books - they are the seller, and therefore the book is coming direct from then and not from an Amazon warehouse. (So that photo full of mail sacks could also have had some Amazon orders in there.)
Simon Edge and Eye Books will therefore know exactly how many books they have sold - either directly from their own website, via their Amazon seller account, via trade orders to bookshops, libraries etc. He/they will know better than Amazon, better than Popbitch (whoever they are) and better than Nielsen.
Just for info, buying via Amazon is totally shit for publishers like Eye Books, if what is happening (as I think it is) i that orders are being fulfilled by Eye Books themselves anyway but via their Amazon store. Amazon take a cut, do not allow any contact between customer and seller (so you've missed out on the nice emails etc) and, if it's anything like where I work, we fulfill orders as quickly as possible (so we aren't punished by Amazon's draconian time rules etc) but without any of the 'frills' a direct customer might get - so we'd definitely prioritise direct customers for signed copies, for example. Independent booksellers and publishers would far, far rather you bought from them than from Amazon, 100%.
Also, Amazon's software/algorithms/whatever... are not to be trusted at all, re bestseller lists, sold out or not, correct photo of item etc.
TL;DR: Simon Edge knows exactly how many copies he's sold; there's no way that anybody else does.