Now that 1.15 is out, I would like to know if I can simply update Minecraft from 1.14.4 to 1.15 without losing my worlds, or do I have to do something in particular?
Should normally be possible
Normally this is of course possible. If you update your Minecraft version, the worlds will be saved and, as far as I know, you will be asked before starting your old world, if you want to play this world with the newest version (Backup to backup is a good idea)
There's a solution how to test this without consequences. You link the world to the "world" folder in Minecraft and give the world a different name. So you have the old 1.14.4 world without damage and if that does not work with the 1.15 a copy.
World folder: Win + R (% appdata% /. Minecraft / saves)
Hope it works and have fun:-)