The first relevant fact that you have to know about young James before I discuss his adventures with the Giant Peach is that he's an orphan. (This is a kids movie so naturally James is an orphan. Parents die more often in kids movies than the token black character dies in a slasher movie.) Apparently, even though his parents lived in London, they managed to get killed by a rhino. Now, London is not generally known for it's roving herds of feral rhinos, but I'll accept that explanation on faith, since the narrator probably knows more about the (possibly circus related?) circumstances of the parents demise at the hands (hooves? horns?) of a rhino than I do. And anyway, how is parents died isn't super important. What is important is that after his parents are out of the way he's remanded to the care of his two terrible aunts.
The old crones are constantly mean to him. So mean, in fact, that young James looks at a giant peach and thinks "well, maybe I can sail this fruit out of here", even though trying to use fruit as a sailing vessel on the ocean is generally not recommended. (The peach does not come with life preservers or life boats.) But with the help of a spider, a grasshopper, and some other talking insects, he does manage to ride the peach into New York City, where he lands it on top of the Empire State Building. (Fortunately, he plopped the peach down while King Kong was off duty. It might not have gone smoothly if their paths had crossed.) James thinks he is home free, but lo and behold, his aunts have driven their cars across the ocean just so they can start being mean to him again. A cop who just happened to be on the scene decides that he doesn't like the look of these mean ladies, so he decides that James doesn't have to live with them anymore. The aunts are banished and James moves into a house made out of the peach's pit in Central Park with the insects as his new family.
This ending is, quite frankly, batshit crazy. Why does a New York cop think that he has jurisdiction over a British family? Has he not heard of "the court system"? Does he not realize that child custody issues are generally solved there, not in the middle of the street? Doesn't he realize that he just sparked an international incident by deciding that a small child from a foreign country would be better off under the care of a talking spider than under the care of his legal guardians? The whole Elian Gonzalez affair became such a major news story and that didn't even involve any of the incredible details that James' case involved, such as a talking spider.
I think the reason why there wasn't a James and the Giant Peach 2 was because the second movie would have to be about his time in a group home waiting for the courts to decide if he would have to be legally remanded back to his aunts, and no one wants to see that movie. (The kids at a group home would probably be less nice to James than the spider was, which is saying something.) Or maybe it's because a movie about James' struggle to get a proper visa into America, even though he crossed the border illegally (and illogically), would be super boring. Or maybe it was because the Giant Peach was mad that it got second billing even though it had to do all the work, so it played salary hardball till the budget rose so high that it wasn't financially feasible to make another one. Or maybe young James ran into the same curse that his parents did, and somehow he found a way to get rhino-ed even though he was living in a major metropolitan city at the time. Come to think of it, there might be a lot of reasons why this didn't get a sequel.