I found a further tidbit; hopefully it will lead somewhere, as it appears his records did not survive.
H. Nash, 29590,Yorks. Regt: along with several other of his mates were admitted on 08 Jan 1917 to the 34th CCS for trench foot. The cover indicates this was at Grove town, (near Somme). On 20 Jan 1917 he was transferred to 1AT.
The first page of this book states there is a diary at the back of the book which must be meticulously kept up as to "the where" of places of transfer, as they moved around so much, but there is no diary in the book. The back cover seem to have a sleeve, so perhaps it was a removable thing.
Have a read-through of
https://blog.nationalarchives.gov.uk/medical-units-first-world-war/ as it might help determine where this 1AT was. The blog gives a link to
https://discovery.nationalarchives.gov.uk/details/r/C4558646 which you can download for free once you register.
Attached is the image of the cover of the book, which covers the dates of 01 Jan 1917 to 12 Jan 1917, in case the numbers on it mean anything.