Despite the religious importance of cutting down a weirwood, many a house across the south kept their weirwoods and godswoods as symbols of status. For example, we see House Tully having kept their weirwood, while at the same time we know they built their castle under Andal rule, and the godswood survived. So, you see that weirwoods weren't only trees that somehow survived the coming of the andals, but they were still planted all across Westeros in later made godswoods, inside castles as a show of status or beauty.
Hell, even the Arryns tried to plant a weirwood in the Eyrie. The Arryns! Which are like, the Highest of the Andal Houses (pun intended), all across Westeros and even have a throne made of a weirwood to show their domination over the First Men.
(Even the f*kin Harren Hoare had a weirwood in Harrenhal, and that also sais a lot. Believe it or not, the Hoares were quite pious to the Faith. One of the first Houses of the ironborn to convert, their Kings were not much liked by the priests of the drowned god for trying to spread the Faith to the Iron Islands. One king even, was named the "Priestkiller", when he had a large number of priests of the drowned god slaughtered when they went against him building a sept in Old Wyk.)
So, you can see how the Weirwoods became more of a symbol of status for the Houses of the South, since having one would be a display of ancientness and wealth, so as to be able to care for it.