It’s a pretty mixed bag.
It seems clear that the various Apostles “planted” churches and appointed overseers (or supervisors) for them.
The people that were left to the task of shepherding the “flocks” and building up the faith (also transmitting the “knowledge” of Christs teaching to future leaders) were disciples (or understudies) of the Apostles themselves. Their “seminary” would have been at the side of their own teacher wherever he went.
The successors of these first bishops were drawn from the local community. Once elected, an Apostle or (later) other bishops of good repute would be summoned to lay hands upon them. If the other bishops did not approve of the choice, they could always refuse. A bishop of a prestigious or senior church could influence the process by witholding approval, and by sometimes calling upon the civil authorities to remove other bishops (that happened a lot after Constantine).
There was no Vatican, no Curia, no Dicasteries, no telegraph, railroad or steamship lines. The bishop of Rome whom would be called Pope eventually could not appoint all of the other bishops as done today. The infrastructure to do that did not exist.
So early on local synods elected the bishops, and delegations of nearby bishops (led by the senior prelate) did the consecrating.
The church of Rome carried a great deal of prestige and was regarded as the school of Peter and Paul, followers of both being present after their martyrdoms. Advice from the major Christian centers was taken seriously, advice and direction from Rome carried more weight than from the other centers.
The church grew from the major cities first, it was an urban religion, and missions were sent out at their expense “up river” to smaller towns and cities. The bishop of the senior metropolitan church in an area eventually would be styled as a Patriarch, or alternatively a Catholicos. The Patriarch of Alexandria was called Pope, just like the Patriarch of Rome.
Eventually, local kings and petty lords (as well as emperors) gained control of the local church institutions and the local synods lost the ability to chose their own bishops. That happened at different times in different places, although it doesn’t seem to have happened in India until the Portuguese arrived. Often when territories and cities would change hands the bishops would be deposed and replaced with countrymen of the conquering king. Control of the pulpit was an important element in pacifying a population.
When William conquered England most of the Anglo-Saxon bishops and many abbots were replaced by Normans, for example.
The ability of the Papacy to appoint all of the bishops did not develop until much later, by fits and starts from country to country. For example, Rome had no such authority in France until it’s Concordat with Napoleon.
I hope that this helps describe the situation.