As far as I can tell…
The Papal Nuncio usually gathers a list of likely candidates. The list usually contains three names listed in order of preference. Then the Pope makes the choice. However, he is not bound to follow the recommendations of the nuncio.
You should also note that it was not always this way. With the advent of modern communication, the Pope has much more knowledge of local affairs that he would have in the past. His ability to be involved in the election of a local bishop is much greater than it had been.
God bless,
Ut