There's no way to magically make a wet, smelly, poopy run stay dry. What your commenters have offered so far are cosmetic fixes. With climate that only occasionally produces wet weather, this is workable. But places like UK that get sloppy weather as a regular feature, you need to do some engineering to create a run that has better drainage.
When people build anything, they often forget to consider "where is the water going to go when it rains?", not just the rain from above but from runoff.
Whether you are situated on flat ground or sloping terrain, you need to see to it that the run and coop are a bit higher than the ground around them or the reverse - trenches. I like to get into my rain gear and stand outside in the rain watching where the water wants to go. I let it tell me what I need to do in order to direct it away from the run and coop.
Rain may soak the run, but with proper drainage, the run will dry out much faster after the rain stops. Even with the deep litter method of coop and run management, you still want to avoid having rain water collect rather than be diverted by means of good drainage around the run.
Also, there's nothing so vile smelling than wet chicken poop. An option is to keep it picked up.