The main reason was that if soldiers were to run, they would have been exhausted by the time they crossed no man's land and made it to the enemy's trench, with little energy left to fight in hand to hand combat. Regardless of whether or not they ran, soldiers were still gunned down by advancing head-on into machine gun fire--running instead of walking does not change this scenario.

Pack weight. First World War soldiers carried an enormous amount of weight on their body. The standard British kit weighed something around 32kg. Though well distributed thanks to a well designed webbing system, that's still a large amount of weight to carry. Add to that the soldiers rifle, helmet, spare ammunition, mills limbs and the weight goes up. Even across short distances that's a lot of weight to try and run with.

In addition to this, in the former years of the war, British infantry needed to advance in a manner that allowed them to maintain formations and groups. Lines running across No Man's Land advancing at a walking pace was the best way to maintain that unit cohesion. During and after 1916, advancing infantry would usually be advancing behind a Creeping Barrage which moved just ahead of the advancing infantry, smothering enemy defences before moving on and allowing the following infantry to capture enemy positions before the enemy had time to recover from the artillery barrage. Creeping Barrage moved according to a predetermined timetable so it was necessary for infantry to keep pace with the Barrage but not get too close so as to take casualties from the artillery shells.

Alway more to be said about this, but you may be interested in the Trench Warfare section of our FAQ.