There was no disaster to tidy up
From purely military perspective, Battle of the Bulge was failed German offensive, and subsequent mostly US counteroffensive. Reasons why German offensive failed are quite simple: at that point of time they were inferior in number of men, number of AFV, number or aircraft and had relative shortage of fuel. It is actually surprising they managed to gather their forces (hard pressed on all fronts) to achieve slight advantage in sector selected for attack. With favorable weather that grounded Allied air support, and an advantage in quality of AFV (arguably in the quality of some units too) , they achieved certain penetration, but their offensive quickly stalled when US reinforcements started coming in and weather improved.
What makes this battle a "disaster" is not things that really happened, but what could have been. This was first and only time in the war that American forces faced large scale German offensive somewhat reminiscent to early war Blitzkrieg victories. Germans had equal or slightly bigger numbers, air advantage was temporarily gone, and some US units suddenly got in the state of shock and paralysis, all too familiar to British, French or Soviets. This didn't last long, it was late 1944, not 1940 or 1941, but it sufficiently shook Us military establishment to understand that US military machine is not invulnerable. What-if scenarios appeared, questioning what could have been if German forces were slightly more numerous and better protected from air . This become even more interesting as Cold War approached, and Soviet armored doctrine was very similar to German.
As for British participation in the battle (especially during German offensive), it was relatively marginal and certainly not decisive. British forces were holding a section of front north from German penetration. Had the German offensive been more successful, they would have to retreat in order to avoid being cut off and left without supplies (Germans were aiming for Antwerp, only major port in that area). Since that didn't happen, British forces simply participated in the counteroffensive with the goal of reducing the bulge created by German offensive.