So what you're talking about as "reserve divisions" isn't actually where replacement men came from to fill battalions. In the case of "reserve battalions" it's a little less clear. But the word "reserve" has a very specific meaning in military parlance and it's less to do with replacements and more to do with adding weight during a battle. Armies did have reserve formations such as divisions, which were to be doled out to a particular corps during a battle to assist in a particular task; taking a line of trenches or holding an important locale. It's the same with smaller units. Reserve battalions were held by a brigade and then tasked to assist other battalions on the battlefield. It's also important to note that units would usually be broken up and fed in piecemeal, as in brigades to assist divisions, battalions to assist brigades and companies to assist battalions.

What you're asking about however, is reinforcements and drafts. These are a completely separate issue to reserves. There were specific units for training up the replacement drafts that came through. Sometimes called "depot" units which didn't have a clearly defined role and sometimes merged a training unit with an operational reserve. But usually each British regiment would have their own dedicated "training battalions" back in the UK or at bases in France. When men volunteered, or were later on conscripted, they would be assigned to a particular regiment for their service. They would then receive their initial training with that regiment before being assigned a specialisation and possibly being moved to a different unit. Once their training was complete these men, who had worked together and bonded as a unit, were sent to France as replacement drafts. Early in the war the battalions were generally responsible for maintaining their own replacement systems which recruited locally. These men would be sent to the various battalions of their parent regiment and serve with others from their region. As the BEF grew this became unwieldy and manpower became more centrally controlled.

Men would train with a particular regiment or battalion in a military district but would became part of the general pool of reinforcements. This split units up and mixed men from different backgrounds (although this wasn't quite true of Scottish, Irish and Welsh units) and regions. This mixed pool would then assign men to particular units as needed with regard only to the ranks needed. It was rare for an entire platoon or section to be replaced entirely and I've never heard of it happening with something as large as a company. More usual was the replacements would be spread out more or less evenly amongst the companies, platoons and sections, hopefully, to ensure some level of widespread experience remained throughout.

There was a lot of disagreement about merging units, particularly battalions, during the war. The bonds of comraderie built up was very strong and the battalion was the sentimental home of most men. It was more important to keep the morale boosting atmosphere of a battalion with its own traditions, than it was to create up to strength units. When the Australian Imperial Force tried to disband battalions in late 1918 there were mutinies against the units destruction that caused a huge amount of upset, including courts martial.