Is it true that Mountbatten was asked by Nehru and Patel to govern the country after independence?

Yes, of course. How else do you think Mountbatten became independent India’s first Governor General? This is unprecedented - a newly independent nation retaining their former colonial leader. And it worked great for India. Once again, an example of the pragmatism that Nehru and Patel practiced.

Around the dawn of Indian independence, India both had British administered regions and 530+ regions administered by local monarchs. The latter regions hoped to get their own independence and stay as independent countries. This is how it looked:

The orange regions were British administered and the blue regions were ruled by local kings. If the blue regions got their own independence, India would have become an ungovernable mess.

To get these 500+ kings to come to line, Nehru & Patel needed a few tricks. They could have used the brute force of the army, but it would have been very expensive, time consuming and unpleasant.

To help them overpower the princes, they needed someone of the stature of Mountbatten. Mountbatten had a “blue blood” - related to all the major royal dynasties of Europe. Indian princes adored such a royal lineage and they found it hard to say “no” when Mountbatten pressurized them to choose between India and Pakistan. Mountbatten also expertly used his political advisor V. P. Menon - who greatly helped Sardar Patel in bringing the Indian princes along the line.

Thus, with very little drama, Mountbatten & his advisor helped Patel bring all but 3 of the princes fall by the line. The last 3 left were Junagadh, Kashmir and Hyderabad.

In Junagadh, Patel took upon the task and got it integrated with pure force. In Kashmir, Mountbatten’s connections in Pakistan helped [both countries retained their British generals] as those Brits could not dream of fighting against their rear admiral and a cousin of the King. The tipoff got by Mountbatten helped India get to Kashmir in time to stop its invasion.

In short, having Mountbatten for a year after independence proved very good for India. Nehru and Patel, having spent long years in jail, did not also have the necessary administrative experience to quickly take over India and needed that extra year to get used to the messy Indian administration and bureaucracy.

On the final kingdom that was holding out - Hyderabad - Mountbatten was dragging his foot. He even offered the option of complete autonomy for Hyderabad province within India [thinking of it now sends shivers - it would have been very messy]. Thus, Nehru and Patel quietly sent him home and brought Rajaji as independent India’s second Governor General. He immediately authorized the use of force and in no time Hyderabad was brought to India.

This pragmatism of India’s leaders is quite underappreciated. They retained Mountbatten when it was favorable to India’s integration and without any drama sent him home when that purpose was satisfied. In 1947, nobody could imagine this massive mess of different administrations could be unified into one system and one constitution. But, it happened.