As recently tested today, compasses do not seem to point towards a spawnpoint when a new one is in place. Compasses only point towards spawn and do not change when the player's spawnpoint changes. This bug occurred in a singleplayer, vanilla, 1.18.1 world. To make sure this wasn't a mistake I made, I made sure to sleep in the bed again before testing this, but the same result occured.

For the images, the first one shows a player character standing nearby a village, the custom made house is where the bed/new spawnpoint resides. The compass is facing away from the bed. The second image shows the bed/new spawnpoint and the compass is facing away.