rmaker:
I'd be interested in hearing more about what you have read regarding resupply.
My own impression of the period is that the men were issued rounds either prior to campaign (Braddock's men were issued 60 rounds per man), or prior to battle.
Unlike most of our wargames, my general impression is that the men didn't shoot as much as we perhaps portray them doing in wargames or even battle re-enactments.
Standard British cartridge boxes (or pouches) carried 18 rounds in the F&I Period. Belly boxes usually carried even fewer rounds (12?). British cartridge boxes in the AWI period carried anything from 28 rounds, to a possible 36 with the Rawles flip block pattern (18 per side of the block).
Again, it is my impression that it is not practical to bring up an ammunition waggon in the middle of a battle to resupply a battalion, as the waggons were usually far back of the lines (such as at Fort Ti, for instance).
As to Sgt. Lamb, I can believe that in the case of Guilford. But just because he refilled his pouch does not mean he was totally out, perhaps just filling in some of the holes with cartridges. He did write that account many years afterwards too, so hard to say.
Most fire fights of the period were short, and usually one side or the other gave way (their morale collapsed). That is what makes Braddock's battle so unique (two and a half hours under fire! Hardly cowards)
What I have always wondered is, where did they keep the additional rounds that weren't in their box? Haversack? Most likely. Knapsack? Not likely. Blanket roll? I doubt that too.
I agree with the above poster. If doing skirmish level, usually running out of ammo is not the worry, as your skirmish, by definition, is probably short.