In WO216/15 an overview of 8th Army units is given, probably dating to mid-October 1941 which I reproduce here. Based on my research, and on the report by GOC 8th Army after the battle (WO201/2693), I have amended it a bit, with explanations given in the notes.
South African Patrol Returning, 21 July 1941, probably Matruh sector. IWM E.4350
Order of Battle
Eighth Army (General Cunningham)
13 Corps (Lt.-General Godwin-Austen)
2 New Zealand Division (Major-General Freyberg)
4 Indian Division (less one brigade)* (Major-General Messervy)
1 Army Tank Brigade (less one regiment)** (Brigadier Watkins)
64, 67, 68 Medium Regiments RA
65, 2 SA AT Regiments RA
A Sqdn 6 SA Armoured Car Regiment
37 RTR (dummy tanks)
30 Armoured Corps (Lt.-General Willoughby-Norrie)
7 Armoured Division (Major-General Gott)***
7 Armoured Brigade (Brigadier Davy)
22 Armoured Brigade (Brigadier Scott-Cockburn)
Support Group (Brigadier Campbell)
4 Armoured Brigade (Brigadier Gatehouse)
1 South African Division (less one brigade)**** (Major-General Brink)
22 Guards Brigade (Brigadier Erskine/Brigadier Mariott)*****
Tobruk Fortress (Major-General Scobie)******
70 Infantry Division (Major-General Scobie)
Polish Brigade Group (Major-General S. Kopanski)
32 Army Tank Brigade (Brigadier Willison)
4 SA Armoured Car Regiment
51, 60 Field Regiment RA
7 Medium Regiment RA
73 AT Regiment RA
Jarabub Group (Force E)
29 Indian Infantry Brigade (Brigadier Mariott/Brigadier Reid)*******
6 L.A.A. Bty. R.A.
1 Squadron 7 S.A. Recce Battalion
2 Squadrons 3 S.A. Armoured Car Rgt.
1 Bty. 2 S.A. Field Regiment
C Bty 73 A.T. Rgt. R.A.
38, “F” RTR (dummy tanks)
2 South African Infantry Division at El Alamein (Major-General de Villiers)
2 South African Infantry Brigade at Matruh Fortress (Brigadier Poole)
849 medium, field, and anti-tank guns
529 cruiser tanks
205 infantry tanks
* 11 Indian Infantry Brigade, held back at Mersa Matruh
** Probably a regiment held back in reserve
*** Detailed breakdown not in the document.
**** The division consisted of 1 and 5 SA Brigades in the line and 2 SA Brigade at Matruh.
***** Brigadier Erskine was replaced by Brigadier Mariott around 20 October 1941, to let Brigadier Reid take charge of 29 Indian Brigade and the Jarabub Group. I have no idea what happened to Brigadier Erskine, since he seems to be a different Erskine to the BGS 13 Corps.
****** This detailed breakdown is not in the WO document, but from me. The fortress came under command 8th Army on the night 30/31 October, and was under operational control 30 Armoured Corps, which would give the order to start the breakout. Later in the battle it went under control 13 Corps.
******* Only HQ 29 Indian Infantry Brigade and one infantry battalion, otherwise support troops including a lot of armoured cars. Under direct control of 8th Army throughout.