Hopefully this may assist you. The following is my take on the structure of the British Army (Infantry) during WWI (in descending order):
Note the Artillery is slightly different
General Head Quarters (GHQ)
So, where the notes on the Battalion (per the info from 1914-1918.net) we see that the 19th (Service) Battalion (4th City) was transferred into the 21st Brigade (which held many Battalions) however it remained in the same Division (which held many Brigades).
I recommend searching for other entries for the 19th (Service) Battalion (4th City) on www.1914-1918.net
. You should be able to work out which Division they were part of.
Sometimes when we can't trace the whereabouts of a Battalion we can look at records pertaining to it's parent Brigade or Division.
I hope my explanation assists.