Hi Kathryn
There is a photo of Adam and his wife Maria on Ancestry public trees by this treeholder: yvonnewilcoxen1 
I have found Adam's WW1 Service Records online and notice something "odd" about one of the children 
Eliza T/Joddeff 30 March 1906 being classed as
 Illigitimate to A.T which would refer to Adam Thornton being the father, then a line is struck through this info.... a bit odd as he and wife Maria Towdoff married Mar Q 1897 @ Lanchester, Co. Durham. Note on the SR his wife's surname is spelt Tordoff.
1911 cen shows a 
Liza Towdoff, niece, 5, born West Stanley interesting 

I cannot find a birth for this person anywhere, have you, just seems rather puzzling.
Just to recapture you stating  
I am surprised they don't seem to have joined the same unit
At the beginning of WW1 most lads/men from same villages/towns/cities etc., did go into the same Battalions which was usually the one most associated with the area they lived in, as the war progressed many gaps were left within Regiments and some Battalions even vanished due to the amount of deaths and casualties, so the others were then put into whatever Regiment needed the men most, in fact my Great Uncle served with the Tyneside Irish, he being a deep rooted Yorkshire man this didn't go down well with him and he lost his life too.
Maybe you have heard of the 
Pals in WW1 these were formed by the men going from same districts into the same Regiments etc., see if you can find the various Pals in the area he lived and see if his name is noted, some were and some weren't.
I have found some Medal Index Cards for a John William/Wm/W Thornton on Ancestry:John William Thornton Lincolnshire Regiment   26459   
John William Thornton Durham Light Infantry, York and Lancaster Regiment, York and Lancaster Regiment   30316, 34621, 34621   
John Wm Thornton 3rd E Lancs FA   354306   
John W Thornton Royal Garrison Artillery   154397   
John W Thornton Royal Engineers   101326   
John W Thornton Duke of Wellington's (West Riding) Regiment, Yorkshire Dragoons (Queen's Own), Royal Engineers, Royal Engineers   4267, 4386, 267307, WR/175663   
John W Thornton Royal Fusiliers, Labour Corps   315877, 405909   
John W Thornton The King's (Liverpool) Regiment, Army Service Corps   28633, T/393668   
John W Thornton Lincolnshire Regiment, Labour Corps   29017, 10060   
John W Thornton Prince of Wales's Own (West Yorkshire) Regiment   5649, 235100   
John W Thornton Prince of Wales's Own (West Yorkshire) Regiment, Royal Army Medical Corps   54039, 146915   
John W Thornton Yorkshire Regiment, Yorkshire Regiment, 4/Yorkshire Regiment   156, 200007, 4381451   
John W Thornton Yorkshire Regiment   4060, 241622   
John W Thornton Welsh Regiment, Machine Gun Corps, Army Service Corps   291092, 143450, M/415272   
John W Thornton Princess Louise's (Argyll and Sutherland Highlanders), Royal Engineers   3118, 148526   
John W Thornton Royal Tank Corps   316557   
John W Thornton  Army Service Corps   DM2/151333   
John W Thornton Royal Army Medical Corps   547, 354306   
John W Thornton Army Ordnance Corps   032676
Remember also he may not given his middle name or initial, just to add extra confusion for you  

Try and find the Service Records for these above and rule out the ones you find that isn't him.
Try and find these in other public trees too, again remember not everyone is diligent at their research so don't take it that any are correct, check yourself ask questions if you have too.
When you have tackled these and not located him, try searching through the dozens and dozens of John Thornton  

Good luck, if you need any more help or pointing in further directions just come back and ask any of us, we will do our best to help.
Regards
bendywendy