So, the idea that the troops who attacked on the First Day on the Somme were ordered to walk comes from a misinterpretation of instructions given to the attacking battalions. The battalions would advance in a series of waves, jumping off into No Man's Land at one-minute intervals and advance at a pace of no more than 100 yards every two minutes (less than two miles per hour). This sounds like the received wisdom that arrogant, out-of-touch Generals believed that the advance would be a walk-over, and indeed, numerous naive junior officers did tell their platoons exactly that (and given the awesome sight of the largest artillery barrage in history, who can blame them?). However, the British planners had very good reasons to order such a pace. Firstly, the attacking troops were carrying up to 80lbs of equipment, which meant that any man running would be rapidly exhausted (entirely necessary equipment as well: the survival of the 36th (Ulster) Division and XIII Corps after they broke into their objective on the First Day was largely determined by the equipment they brought with them to reverse the captured German trenches). Secondly, the BEF's rapid expansion was such that there was a terrible lack of experienced officers and NCOs who were capable of conducting more complex tactics, even though their men were entirely capable. Third, it was felt that the inexperience of the battalions could be compensated for by the power of the artillery barrage, which would smash dugouts, cut wire, and stun the Germans into insensibility before the first wave advanced (again, not an entirely misplaced idea: exactly this happened in the Ulsters' and XIII Corps' sectors).

A steady pace makes even more sense when we examine the "Wave System" the battalions advanced in: the infantry battalion of four companies advanced with the two leading companies intended to sweep right to the back of the enemy first line and prepare to receive the counterattack. They each advanced with two fighting platoons up, and two fighting platoons just behind, with an interval of a hundred yards between the platoons. Behind those two companies came the third company, with two platoons assigned to "mopping up" duty (to secure the trenches behind the lead fighting platoons) and two platoons assigned to support duties. Behind that came the battalion HQ, the last company assigned to carry forward equipment, and finally stretcher bearers. In all, the battalion advanced in ten waves, with five yards between each man and 100 yards between each wave, covering a frontage of 400 yards with a depth of 900 yards. You can see why a slow pace would be wanted: the alternative would be the waves becoming strung out or mixed together.

In conclusion, a steady advance was mandated by the circumstances the British faced on the Somme, including the inexperience of their battalions, the amount of equipment they had to carry, and a (ultimately misplaced, but hardly unreasonable) belief that the artillery barrage was so powerful that they would not be contacted until they had broken into the first line of trenches. In several places on the Somme, most notably the Ulsters at the Schwaben Redoubt and XIII Corps at Montauban and Caterpillar Valley, it actually succeeded. Had XIII Corps' successes been exploited, the British may well have gained the German second position on the First Day and redeemed their casualties a little. Alas, with Rawlinson in command, that was never going to happen.

Sources:

Gordon Corrigan, Mud, Blood and Poppycock

Martin Middlebrook, The First Day on the Somme

William Philpott, Bloody Victory: The Sacrifice on the Somme

Gary Sheffield, The Chief: Douglas Haig and the British Army