[Selmo]

We know that the population of The Shire increased rapidly after the War. leading to its expansion westwards into the Tower Hills. There's no reason to think that the same thing would not happen in the rest of Middle Earth.

Within Aragorn's lifetime, Gondor could recover much of its former strength. I don't think that the Kingdoms in the East would be much of a treat to Gondor. Many of their leaders died in the War, the people would no longer have a common purpose, there would be power struggles between new aspiring leaders and it would be a very long time before any one leader could become stong enough to command great armies. Without Sauron's promptings, would they have any incentive to risk another western adventure?

The powerbase for Aragon and his line in the Southern Kingdom would be secure for many generations.

Is the United Kingdom of Gondor and Eriador just a paper dream? No.
There hasn't been a central government in the North for a thousand years but the idea of the Kingdom has remained alive in the hearts and minds of the Dunadain in their hidden valleys, in the tales the Hobbits in The Shire and even among the people of Bree. With the onset of peace and a rapidly increasing populaton, these groups, together with other smaller settlements of hobbits and men that Tolkien only hinted at, could be pulled together into a Kingdom, further strengthened by immigrants from Gondor.

How long would the United Kingdom last? Not for ever.
After the War of the Last Alliance there was a time of peace and expansion but, because of the distance between the two, there was a need for sepparate administrations in Gondor and Eriador . Gradually, the links between the North and the South disolved, not through the actions of external forces but simply because of the difficulties of communication. The same thing would happen again.