Did the British really bail out the Americans at the Battle of the Bulge?

During the battle of the bulge, north side of the german salient was given to Montgomery command, because for a while Eisenhower thought it would be easier to organize the battle in that way. Because american 12 group army (1st and 3rd american armies) headquarters was in the south of the salient, while 21 group army (1st canadian, 2nd british and 9th american armies) was in the north of it.
Therefore, Montgomery had to take command of 1st american army for a while, but not to help United States to control a retreat or a desperate situation, but instead it was just a tactical decision.

Source: 21st Army Group

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.

The origin to the idea that Montgomery claimed to have rescued an incompetent US command at the time of the Ardennes offensive, is almost entirely due to a cleverly produced piece of German propaganda that was put out in the guise of a BBC announcement. Designed, as it obviously had been, to drive a wedge between the two western Allies, it was clearly successful - as witnessed by the fact that writers such as the one quoted in the question are still, 78 years later, confused about the facts.

You can read all about it in this essay (2020) by staff of the National Archives in Britain.

Initially all concerned - including American forces all over Europe - fell for what appeared to be a BBC report. Unfortunately the matter was taken up by some of the more jingoistic press in Britain - notably the Daily Mail - for whom it wouldn't have taken its editor much to fall for such a piece of "fake news".

From what I have read elsewhere, there may be some evidence that Montgomery warned Eisenhower, a couple of weeks before the German offensive, that it was likely to happen due to the large gap in the Ardennes front, held only by troops that were "resting". And he may already have urged Eisenhower to move Patton's forces north at that time.

Remember the British commanders had had greater experience of battle - from WW1 - than their American colleagues had had. But I strongly suspect that Monty, having previously not withheld his contempt of some of the US's less-experienced commanders, ( especially in Italy) was not taken as seriously as he should have been on this occasion. This was surely an important lesson for both sides.

As for the Ardennes battle, the facts of the numbers killed, wounded and missing must leave in no doubt the question of who bore the brunt of the German offensive. American: killed 8,407, missing 20,905, wounded 46,170 British killed 200, wounded 969, missing 239 (Wikipedia, quoting SHAEF original numbers). But let's not overlook the larger number of Germans - killed 10,749; wounded 34,225, captured 22,483 - mostly, without doubt, ordinary decent chaps, but the victims of a frantic last gamble by the maniac who governed them.