The source of information for this is gorynych forum, a site devoted to the history of the Urals Cossacks.
It aligns in most respects with the outline of the army given in other sources, although many regiments are cited by different names.
1st Ural Cossack Division
1st Ural Training Regiment – 6 sotnias, 24 officers, 520 sabres, 14 MGs
3rd Ural Training Regiment – 6 sotnias, 20 officers, 540 sabres, 7 MGs
Guriev Cavalry Regiment – 4 sotnias, 12 officers, 200 sabres, 6 MGs
Kalmykov Cavalry Regiment – 4 sotnias, 13 officers, 270 sabres, 5 MGs
Guriev Foot Regiment – 4 companies, 12 officers, 630 bayonets, 12 MGs
1st Ural Horse Artillery Divizion – 7 officers, 6 guns, 1 howitzer
2nd Ural Cossack Division
Lbischensk Cavalry Regiment – 4 sotnias, 8 officers, 400 sabres, 10 MGs
Uralsk Cavalry Regiment – 4 sotnias, 11 officers, 325 sabres, 9 MGs
Sakharnovsk Cavalry Regiment – 4 sotnias, 12 officers, 240 sabres, 6 MGs
1st Partisan Cavalry Regiment – 4 sotnias, 11 officers, 560 sabres, 14 MGs
2nd Partisan Cavalry Regiment – 4 sotnias, 10 officers, 300 sabres, 6 MGs
Uralsk Foot Regiment – 4 companies, 12 officers, 300 bayonets, 7 MGs
2nd Ural Horse Artillery Divizion – 7 officers, 8 guns, 2 howitzers
4th Iletsk Cossack Division
2nd Ural Training Regiment – reforming
8th Cavalry Regiment – 6 sotnias, 5 officers, 405 sabres, 10 MGs
16th Cavalry Regiment – 4 sotnias, 6 officers, 850 sabres, 11 MGs (text has 15th Regiment)
Composite Cavalry Regiment – 4 sotnias, 11 officers, 780 sabres, 6 MGs
Iletsk Foot Regiment – 4 companies, 10 officers, 230 bayonets, 8 MGs
Separate Burenin Brigade
Burenin Cavalry Regiment – 6 sotnias, 8 officers, 950 sabres, 14 MGs
Irtetsk Cavalry Regiment – 3 sotnias, 5 officers, 420 sabres, 6 MGs
Sakmarkin Partisan Sotnia – 1 officer, 420 sabres, 6 MGs
Sal'nikov Partisan Detachment – 4 sotnias, 11 officers, 385 sabres, 4 MGs
Northern Partisan Detachment – 1 sotnia, 1 officer, 130 sabres, 6 MGs
7th Horse Battery South – forming, 2 guns
3rd Ural Cossack Division
Rubezhensk Cavalry Regiment – 4 sotnias, 7 officers, 5-0 sabres, 7 MGs
Kirsanovsk Cavalry Regiment – 4 sotnias, 8 officers, 610 sabres, 6 MGs
2nd Composite Foot Regiment – 3 companies, 2 officers, 260 bayonets, 5 MGs
4th Mounted Battery – 3 officers, 2 guns
Slamikhinsk Detachment
Slamikhinsk Cavalry Divizion – 3 sotnias, 9 officers, 310 sabres, 5 MGs
Chizhinsk Partisan Divizion – 3 sotnias, 7 officers, 540 sabres, 25 bayonets, 10 MGs
Steppe Partisan Detachment – 5 officers, 100 sabres, 325 bayonets, 8 MGs
Novouzensk Regiment – 4 sotnias, 13 officers, 215 sabres, 4 MGs
Al-Gaisk Partisan Detachment – 2 sotnias, 4 officers, 220 sabres, 3 MGs
Savage Partisan Detachment – 1 sotnia, 4 officers, 70 sabres, 4 MGs (quite likely Kirghiz)
6th Horse Battery – 5 officers, 4 guns
Guriev Brigade
Guriev Infantry Divizion – 2 companies, 16 officers, 425 bayonets, 8 MGs, 2 guns (37mm)
Astrakhan Infantry Divizion – 6 companies, 30 officers, 1,915 bayonets, 14 MGs, 4 guns
Special Sotnia – 1 officer, 125 sabres, 1 MG
33rd Nicholaevsk Infantry Regiment – 1 battalion, 9 officers, 630 bayonets, 12 MGs
Volunteer Turkestan Battalion – 3 companies, 6 officers, 260 bayonets, 2 MGs
Armoured Car Divizion – 26 officers, 30 bayonets, 7 MGs, 2 guns
"Crocodile", "Hedgehog", "Lazarus" and "Meringue"
Separate Temir-Khan-Shura Sotnia – 20 officers, 62 bayonets, 2 MGs (Lewis) (uncertain if here)
Separate MG Sotnia – 11 officers, 30 bayonets, 6 MGs
1st Muslim Battalion – forming
2nd Muslim Battalion – forming
1st Horse-Mountain Artillery Divizion – forming, 2 guns
The total listed is 419 officers, 10,415 sabres, 5,120 bayonets, 265 MGs and 35 guns.
These numbers are for actual men with rifles or sabres actively fighting in the formation. As such they grossly underestimate the actual strength of the army – for example even front-line gunners in the artillery units are excluded.
This list appears to leave out any units part of another host, so the 13th Orenburg Cossack Regiment is not listed.
The strength of the Astrakhan Infantry Divizion appears seriously wrong. Firstly it is extra-ordinarily large. All other sources give a reasonable sized infantry unit (often designating it Astrakhan Plastoon Regiment), but also list an Astrakhan Cossack Regiment and another non-Urals Cavalry Regiment fighting alongside it.