How long does it take for water to naturally heat in a pool to an acceptable degree for swimming when you first fill your pool?

Actually depending on the weather, the pool can heat up in a few hours.

It depends on the depth of the pool, the contours, length of sun exposure, and latitude. For example, a round kiddie pool 8 inches deep and 4 feet in diameter heats up to 90 degrees F. in about 2 hours in full sun. From this example, if you want to, you can figure out the rough dimensions of your pool and the volume of water in it with simple math.

Anecdotally, it takes my backyard/in-ground pool about 1 week to heat up to swimming temps (75+ degrees) providing the highs remain at 90 degrees F. or better and the night time temps don't fall below 60 degrees.