tl;dr - Almost nothing changed in Montgomery's situation, he continued in command of the 21st Army Group which included most (all?) of the Commonwealth combat divisions, and several US units, until the end of the war.

For a while since the invasion of Normandy Montgomery had been pressing for Eisenhower to appoint a single commander for the ground troops in NW Europe. Naturally his intent that this commander would be himself. Remember that Montgomery was the ground forces commander for Operation Overlord, having command over Bradley and US 1st Army, until the command structure was changed in July 1944. He also wanted all the allies effort focused on a single push into Germany. Again his intent was he would command this push. Eisenhower resisted these suggestions, not least because the US was already providing the bulk of the armed forces in Europe, but found it difficult to argue against Montgomery's charisma and experience.

Following Market Garden, Montgomery continued to argue for a single commander and a single push but the over-worked Eisenhower was now determined to stick to the plan of advancing on multiple fronts. The relationship between the two became further strained and Montgomery wrote an ill-advised letter to Eisenhower which almost cost him his job in October. Eisenhower was forced to give Montgomery a direct order to open him the port of Antwerp to shipping in order to relieve the Allies logistics problems.

When the Germans launched their winter offensive (The Battle of the Bulge) on December 16th 1944 they quickly drove a wedge between the US 1st and 9th Armies on the north side of the bulge and the rest of the US 12th Army (see 1 below). Bradley's HQ was south of the Bulge and in order to simplify communications control of those Armies was temporarily transferred to Montgomery. Following the reduction of the pocket control was returned to Bradley, but the US commanders did not appreciate the interference and liked even less Montgomery's implication to the press that he alone had saved the Americans.

However, Montgomery's reputation in the UK was such that it was pretty much unaffected by the failure of Market Garden. After the war he commanded the British forces occupying Northern Germany, which became the British Army Of the Rhine (BAOR), was appointed as Chief of the Imperial General Staff and also served as Deputy Supreme Commander of NATO.

Some other points:

1. Montgomery was just one of three ground forces commanders under Eisenhower from August 1944 to the end of the war in Europe. The other two were Bradley, commanding the 12th US Army Group and Gen Jacob Devers, commanding the 6th Army Group^*. Patton commanded the 3rd US Army under Bradley (mostly).

2. Although Market Garden failed to achieve its ultimate goal of seizing the Arnhem bridge, the ground that was taken during the operation was crucial as a jumping off point when the offensive resumed in early 1945.

3. Patton would not have been appointed to command the 21st Army Group which, as noted, was largely composed of Commonwealth troops. Such a move would have been politically impossible.

principle source : Rick Atkinson, The Guns at Last Light

^* Devers command responsibilities were very complex and fluid and well outside the scope of this question.