Is it not disrespectful of the people of the North for the King's Landing godswood not to have a weirdwood tree? I understand that there was some cutting-down of weirwoods south of the neck, but hadn't Faith-of-Seven-vs-old-gods conflicts died down by the time of Aegon I's landing?
tl;dr: There's nothing preventing King's Landing from regrowing a weirwood tree, but I doubt anyone in charge there is motivated enough to bother.
It's true that by the time of the series beginning, the Andals had made peace with the First Men, including a (grudging) acceptance of their religion. However, by that time, the damage had been done.
When the Andals invaded, they cut down almost every weirwood tree they came across, without fail. As far as we know, there are only a handful of trees left outside the North (Casterly Rock, Riverrun, and Storm's End, for example, plus a grove on the Isle of Faces). This means any weirwood trees in or around King's Landing were destroyed.
To replace it would require someone to intentionally plant a new tree, and we're led to believe that the trees are very old. So it would take years, possibly decades, before a new weirwood tree was in a condition to be carved up and used religiously. But King's Landing, while it may tolerate the Old Gods, is primarily populated, and run, by worshippers of The Seven. It's unlikely that anyone with both the authority and the desire to try to replace the weirwood tree even lives in King's Landing anymore.
The central focus of a Godswood is the heart tree. Typically, this should be a weirwood, but as you mention all the weirwoods south of the Neck we cut down.
In the south the last weirwoods had been cut down or burned out a thousand years ago, except on the Isle of Faces where the green men kept their silent watch. Up here it was different. Here every castle had its godswood, and every godswood had its heart tree, and every heart tree its face.
A Game of Thrones - Catelyn I
However, the heart tree does not have to be a weirwood to have the same purpose. A you mention the Kings Landing Godswood does not have a weirwood, but it still has a heart tree.
The heart tree there was a great oak, its ancient limbs overgrown with smokeberry vines; they knelt before it to offer their thanksgiving, as if it had been a weirwood.
A Game of Thrones - Eddard V
Which seems to be enough for Ned to feel the comfort of the gods he prays to
The heart tree was an oak, brown and faceless, yet Ned Stark still felt the presence of his gods. His leg did not seem to hurt so much.
A Game of Thrones - Eddard XII
So to conclude while every Godswood has its heart tree, but that heart tree is not 100% a weirwood.