There was a lot of training involved for D-Day. A huge difference between the WWI and WWII (or at least D-Day), is that WWI was already on the mainland and under threat. Reinforcements were needed quickly. Before D-Day, the Western Allies had the time to deliberate and prepare for their great offensive on the mainland.

Another part is that as time went on, the conscripts and volunteers may not be any better in skill or training, but the officer corps was strengthened and training techniques were honed. The run of the mill conscript didn't really need to know complex tactics, all they needed to do was follow orders from the officers who did.

Probably the biggest change was in communication. In 1944, at Normandy, everyone had radios. Every company had one. In the US every tank and airplane had one. This meant every discrete unit on a battlefield could act as an observer and provide realtime info to their commanders.

So a captain could communicate to his headquarters both his and the enemy's positions. His commander could then issue detailed commands back to his individual companies. Or he could call up and ask for artillery or air support with a high degree of confidence that either would hit the enemy and not his own men. This allowed a degree of flexibility and mobility that was unknown in WWI. Individual companies could be given a wide latitude of independence without (as much) concern of where everyone was, and if everyone was acting in a cohesive whole, because commanders knew more or less where each company was and what each company was doing.

My understanding is that Haig and other generals didn't believe the newly trained men of Kitchener's Army (volunteers rather than conscripts, but still not professionals) could handle anything more complex than walking slowly forward across No-Man's-Land and thus developed the plan of annihilating the German defences with artillery first, so that was all they would need to do.

It wasn't that the troops weren't trusted. In fact combined arms tactics were largely invented during WWI and by 1918 the British conscripts were acting far more like their WWII equivalents than the 1914 army. And even in 1914 they had a degree of flexibility that was lost once the front solidified into trench warfare.

The first problem, was artillery proved to be decisive, even in a way that wasn't true in the 19th century (when it was already considered the decisive arm). Infantry unsupported by artillery always lost to infantry that was supported by artillery. And this is where communications come back in. The defensive party (from 1915-1918 usually the Germans) had communications from the front lines back to their artillery. They had buried phone lines. The offensive party would generally lose their communications to their artillery within a few minutes of the start of an offense. Coordination with the infantry on the fly was impossible. And moving infantry without artillery support was suicide (of course moving with artillery support wasn't much better). While the defending army could order artillery strikes on the fly at the advancing men.

So the way they tried to work around this was to have a plan. The artillery would fire at specific times and places. And then the infantry would advance and hit the area the artillery had just hammered. Infantry had to move at precise moments to precise areas, or when their artillery opened up again it would be hitting their own troops. Communication would be unnecessary because everyone would just be where they were supposed to be according to the plan. Of course in real battle nothing would quite go according to plan. Everything would be a royal mess pretty quickly. And planning on this level meant moving large bodies of men, because planning moves on a company by company level was beyond the planners in 1915. This "Bite and Hold" model of combat was the dominant model on all sides from late 1914 to 1918. The artillery would bite an area; then the infantry would move in, as a mass, finish off any survivors and hold it. Of course the biting never was as decisive as hoped and holding was always a difficult prospect. The only certain thing was that a lot of men would die.

By 1918 this had changed a great deal. The artillery, while still not responsive in real time, had developed a finer degree of control. And they had developed new tactics like rolling barrages. Instead of hammering at an area for hours, the artillery would fire for a relative short period of time in a wave. And the infantry instead of waiting for the barrage to end, would advance right behind the wave of where the shells were landing. This allowed them to be in the German trenches before the Germans could get back to their positions. Tanks were first invented, then improved upon. This allowed the infantry to have mobile local artillery and machine guns, a kind of close fire support artillery that was missing in the early part of the war. Airplanes began to carry radios and could act as artillery spotters on the fly.

And once the Hundred Days offensive began in 1918, the Allied armies looked and acted a lot more like their WWII equivalents. The communications back to headquarters still didn't quite exist yet. But with some planes and tanks having radios there was some degree of communication. A company commander could knock on a tank and ask them to take care of a particular obstinate position. And the few radios meant artillery could be redirected instead of just going by the plan. As the war turned back into a mobile one it also turned back into a fight of companies against companies, instead of divisions against divisions. When that happened local captains and their non-commissioned officers proved to be plenty well trained for small independent actions.

Fast forward to 101 years later and the Battle of the Somme. My understanding is that Haig and other generals didn't believe the newly trained men of Kitchener's Army (volunteers rather than conscripts, but still not professionals) could handle anything more complex than walking slowly forward across No-Man's-Land and thus developed the plan of annihilating the German defences with artillery first, so that was all they would need to do.

I think you're putting the cart before the horse here; it was because of Rawlinson's faith in the artillery bombardment that the infantry were 'expected' to 'walk' through. It was the infantry tactics being tailored to fit the artillery, not the artillery tailored to fit the infantry.

Moreover, it needs to be said that the BEF training manual at the time, SS 109 (based off French manuals) involved fairly complex maneuvers, and assigning specific roles like 'trench sweepers'. The only problem was that the BEF training apparatus was poorly prepared for the massive expansion of the BEF, though it would be vastly improved by 1917, when manual SS 143 was implemented. The divisions arrayed on July 1st also lacked a core of combat-experienced troops, meaning that they lacked substance and were poorly prepared for action.

It should, however, be noted that the idea that all the battalions 'walked' on July 1st is a myth. Robin Prior and Trevor Wilson have debunked this pretty heavily, with the evidence suggesting that c. 50 of the 80 battalions that went over the top were already in no-Man's land at 07:30, c. 20 ran from the parapet, and the last c. 10 walked, though many of these were behind creeping barrages, and enjoyed great success. Ivor Maxse was able to put his 18th Division through a serious training regimen, that left them better prepared than most for battle.

So, now it's 1944 and Britain is preparing for D-Day. Their conscripted troops are now considered able to participate in complex combined ops.

The allies had rehearsed and meticulously prepared for operation Overlord for months and years before hand, since 1943 at least. Overlord itself was 2 years in the making, while the Somme Offensive went through numerous 'drafts', and was planned from about March 1916 onwards.

One thing not yet mentioned here is an increase in literacy. For most of history, the grand majority of soldiers were not literate, but by the 20th century, most people were capable of some level of reading. This, combined with industrialized printing and improvements in communication like the radio and telegraph, allowed armies to mass produce instruction manuals and battle plans that could widely and efficiently shared.