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