I think this is really a tricky one , and depends entirely on the Officer's own choice , and where he was operating .I don't have a definite answer , and I don't think there's likely to be a hard and fast rule ; the " Special Forces " would make up their own rules anyway , but in the Line Infantry the situation is unclear .
We need immediately to distinguish between the Junior Officers , up the Sharp End with the troops , and the Senior Officers , who were just that little bit further back at the Battalion HQ , who were mostly ( but not always ) out of the firing line , and probably wore just a pistol.
I get the impression ( from pics and memoirs ) that the full set of Officer's webbing was rarely worn by anyone : they wore what was most useful at the moment : and since many officers had use of a vehicle , they tended to leave spare items of kit in it until needed : this was probably true of most Captains and above.
The Lieutenants , actually marching with their platoons , would choose their equipment according to their judgement as to whether it would make them conspicuous : some of them ( with very little judgement or experience ) probably stuck out horribly , wearing obvious Officer's webbing items , and carrying pistols , though I expect many would have carried stens or rifles as well if they had any experience .
The most junior officers would probably look more like NCOs or ORs in the field , down to wearing OR's haversacks on their backs at times. I expect pips were always worn : they are very inconspicuous in combat , having been invented during the Great War entirely because the previous cuff ranking was so visible .
The cleverer ones probably wore ORs webbing , at least partially , with other items added as needed : they would certainly need their binoculars , a compass , and possibly a map case , as well as the obvious neccessities like a waterbottle and a haversack to put their rations in .
I think the best place to look is in memoirs , where you can sometimes pick up little details : Peter White , the junior KOSB Officer I've made a figure of , states in his book ( " With the Jocks " ) that he always carried a rifle : so presumably he also carried a pair of universal pouches to carry ammo and grenades ; but he also mentions binoculars and maps and compasses , and he must have had somewhere to put them ....
To answer your ( entirely pertinent ) question , I think it all depends....
And it's odd that no-one ever asks this question about the Germans !
*************************************************************