Why did Lord Mountbatten shrink the timeline for the withdrawal of British troops from India?

This is an extremely contested question!

By 1947, UK policy was to "quit India" on an aggressive timetable, even though many aspects of its future governance were not established - in particular, whether it would be partitioned. In the Parliamentary statement of 20 February 1947 announcing Mountbatten's appointment as Viceroy, the Prime Minister Clement Attlee also outlined a policy where the British departure would take priority over answering those questions -

His Majesty's Government desire to hand over their responsibility to authorities established by a Constitution approved by all Parties in India in accordance with the Cabinet Mission's plan, but unfortunately there is at present no clear prospect that such a Constitution and such authorities will emerge. The present state of uncertainty is fraught with danger and cannot be indefinitely prolonged. His Majesty's Government wish to make it clear that it is their definite intention to take the necessary steps to effect the transference of power into responsible Indian hands by a date not later than June, 1948. [...] But if it should appear that such a Constitution will not have been worked out by a fully representative Assembly before the time mentioned, [...] His Majesty's Government will have to consider to whom the powers of the Central Government in British India should be handed over, on the same date, whether as a whole to some form of Central Government for British India or in some areas to the existing Provincial Governments, or in such other way as may seem most reasonable and in the best interests of the Indian people.

Mountbatten's official biography by Philip Ziegler (1985) summarizes his private instructions:

Keep India united if you can; if not, try to save something from the wreck; whatever happens, get Britain out was the essence of Mountbatten's orders.

Interviewed by Dominique Lapierre and Larry Collins for their book Freedom at Midnight (1975), he said:

We couldn't have gone more slowly [...] The Interim government which I inherited and didn't create, which had, from memory, 6 Congress, 3 minorities, that's 9, against 5 Muslim League - was a government which could not operate [....] When I named August 15, I admit I chose the anniversary of the surrender of the Japanese in Singapore but I had to name a date, and it was a good date. But, and this is the point I really want to make, it couldn't have lasted longer.

(To be precise, 15 August 1945 was the date of the surrender of Japan in general, and the Singapore surrender ceremony at which Mountbatten presided was not until 12 September.)

His stated perspective was that he could not hold the existing government together until the original date, and so decided to bring the transition forward in order to force the issue. Subsequently, he was successful in integrating the princely states, but failed (for whatever reason) to keep India and Pakistan together. The fast timetable also meant that the line of partition was famously decided hastily and in fraught circumstances, causing immediate misery and lasting conflict.

This was arguably in line with Attlee's policy, and the Viceroy had an immense concentration of power that could not be checked by any authority within British India. Mountbatten certainly put about the version of the story where he - unlike officials or Parliamentarians in London - uniquely realized the gravity of the situation in India and acted within his discretion.

On the contrary side, and from critics both in India/Pakistan and in the UK, it was suggested that his decisions were not only mistaken, but motivated by a personal desire to extricate his career from association with disaster. He remained as Governor-General of India until the original suggested deadline of June 1948 (but did not take that office in Pakistan), whereas in the February plan he might plausibly have retained more authority, as Viceroy, in a longer transition, and therefore have carried more responsibility for its success or failure. As it happened, despite bitter criticism, he was able to continue a distinguished naval career: rising from two-star to five-star rank, becoming Chief of the Defence Staff, and receiving various honours from the British state.